Research on Sound Absorption Properties of Tri-Periodic Minimal Surface Sandwich Structure of Selective Laser Melting Titanium Alloy

Abstract

Triply Periodic Minimal Surface (TPMS) sandwich structure has the characteristics of lightweight, high specific strength, specific stiffness, vibration, and noise reduction. There is relatively little research on its sound absorption performance. TPMS sandwich structure based on selective laser melting (SLM) technology can achieve precise control of structure type, porosity, and so on, and has broad prospects in noise control applications. In this paper, the sandwich structure is designed based on tri-periodic minimal surface implicit function, and the titanium alloy sandwich structure is formed by SLM technology, and the sound absorption performance of titanium alloy TPMS sandwich structure is studied. The effects of volume fraction, panel thickness, and cell layer number on the sound absorption performance of the two TPMS structures were systematically analyzed using the transfer function method. The results show that: The GP10 sandwich structure with the volume fraction of 20%, the thickness of the panel is 1.0 mm, and the number of cell layers is 3C parameter combination has better sound absorption performance and higher sound absorption bandwidth, and the sound absorption coefficient is 0.36. For Ti6Al4V sandwich sound absorption structure, it is not suitable to design too thick panels. When the volume fraction is lesser, the increase of the tortuosity factor τ improves the sound absorption performance of the two structures, and excessive volume fraction leads to a decrease of sound absorption performance. The increase of cell layers can broaden the sound absorption bandwidth of the two structures, and the sound absorption capacity increases and then decreases in the range of 150–6400 Hz. The Gyroid structure mostly exhibits resonance sound absorption mechanism, while the Diamond structure exhibits resonance sound absorption mechanism combined with the viscous loss of sound wave. The work done in this paper can provide a theoretical basis for the study of the sound absorption characteristics of TPMS sandwich structure.

1. Introduction

Sandwich structure is a multifunctional composite material, with heat dissipation,¹⁾ impact resistance,^2,3) vibration and noise reduction,^4,5) and other characteristics. Sandwich structure types mainly include foam sandwich structure,⁶⁾ honeycomb sandwich structure,⁷⁾ lattice sandwich structure⁸⁾ and so on. Among them, honeycomb sandwich structures and foam sandwich structures have been widely used in rail transit,⁹⁾ aerospace,¹⁰⁾ and other fields, while lattice sandwich structure is considered as the most promising new lightweight high-strength structure which is gradually being widely used. H Meng¹¹⁾ studied the acoustic performance of the honeycomb and corrugated plate composite sandwich structure with micro holes, and the results showed that the composite structure with micro holes showed higher sound absorption performance at low frequency, and the heterogeneously perforated composite structure had larger sound absorption bandwidth. Wang Da¹²⁾ found that the size of the cell in the honeycomb structure not only affects the first and second resonant frequencies but also their peaks, in addition, he also optimized the sandwich structure by genetic algorithm to make it perform well in train noise control. Tang YF¹³⁾ et al. established a theoretical model of acoustic impedance based on microperforated plates by combining a honeycomb sandwich and a lattice sandwich. Through the study of panel thickness, corrugated sandwich thickness, and core height parameters, they found that thicker panels and higher core heights could enhance sound absorption performance in the low-frequency range. Wang¹⁴⁾ used the theory of sound-electric analogy, divided the microporous face-centered cubic structure into several series and parallel sound absorbers to study the sound absorption performance, found that the premise of resonant frequency moving to low frequency is to reduce the number of parallel sound absorbers, and increase the number of series sound absorbers will increase the number of formants in the absorption curve and decrease the first resonance frequency; At the same time, Wang used the simulated annealing algorithm to optimize the relevant parameters of the lattice sandwich to obtain a wider sound absorption band. Compared with the traditional sandwich structure, this new lattice sandwich structure has higher strength and sound absorption performance.

It can be seen that the design of the core of the sandwich structure affects the acoustic and mechanical properties of the sandwich structure. Traditional metal sandwich has certain advantages as sound-absorbing structure, but most of them are connected by periodic trusses,^15–17) which leads to stress concentration and damage at the intercellular nodes, resulting in the degradation of mechanical properties of the structures. A triply periodic minimal surface (TPMS) is a kind of surface with a smooth surface and highly connected holes. The structure is precisely controlled by implicit function, which improves the design efficiency of a sandwich structure. As a typical additive manufacturing technology, selective laser melting (SLM) takes a laser beam as the heat source and scans layer-by-layer on the metal powder bed according to the path planning in the 3D slice model. The scanned metal powder melts and solidifies rapidly and forms parts after layer-by-layer accumulation. It overcomes the difficulty of forming metal parts with complex structures by traditional technology and provides an effective means for the accurate design of the sandwich structure.

At present, the acoustic performance of TPMS sandwich structures formed based on 3D printing technology is still in its infancy. Diab W¹⁸⁾ has studied the acoustic attenuation and band gap of TPMS-based structures. The results show that the Gyroid structure has acoustic attenuation ability and can obtain a wide band gap at low porosity. Sun et al. prepared 316L stainless steel porous structure using SLM technology to study its sound absorption performance, and the results showed that the variation of unit structure affects its sound acoustic performance, and the sound absorption performance of 316L porous structure was better than that of solid structure in the measurement frequency range (1–6.3 kHz).¹⁹⁾ Yang W J et al. studied the acoustic performance of Diamond, Gyroid, and Primitive polymer structures, and the results showed that among the three structures, Diamond structure showed excellent absorption capacity in a wide bandwidth.²⁰⁾

The researches on triply minimal surface sandwich structures mainly focus on statics and dynamics, the core is also widely used in the field of bioengineering.^21–23) However, there are few researches on their acoustic properties. Based on the above, Gyroid and Diamond structures with good sound absorption performance are selected in this paper. The sound absorption performance of SLM-formed titanium alloy TPMS sandwich structure was experimentally investigated. The influence of volume fraction (V), panel thickness (P), cell layer number (H), and structure type on the sound absorption performance of TPMS sandwich structure was systematically analyzed, which provided the engineering application basis for the research of TPMS sandwich structure.

2. Design and Experimental Methods

2.1 TPMS sandwich structure design

The sandwich structure of TPMS is composed of “solid panel -TPMS porous structure - solid panel”. The surface in a cube of TPMS porous structure can be expressed as:   

\begin{align} G(x,y,z) &= \cos(\alpha x)\cdot \sin(\beta y) + \cos(\beta y) \cdot \sin(\gamma z) \\ &\quad + \cos(\gamma z) \cdot \sin(\alpha x) + t \end{align} (1)

  

\begin{align} D(x,y,z) & = \sin(\alpha x) \cdot \sin (\beta y) \cdot \sin(\gamma z) \\ &\quad + \sin(\alpha x) \cdot \cos(\beta y) \cdot \cos(\gamma z) \\ &\quad+ \cos(\alpha x) \cdot \sin(\beta y) \cdot \cos(\gamma z) \\ &\quad + \cos(\beta y) \cdot \sin(\gamma z) \cdot \cos(\gamma z)\\ &\quad + \cos(\alpha x) \cdot \cos(\beta y) \cdot \cos(\gamma z) + t \end{align} (2)

Where α, β, and γ control the unit size of a single cell in X, Y, and Z directions, period T = 2π/α, t is the relative density control parameter. Equations (1) and (2) are the governing equations of Gyroid and Diamond surfaces. Taking the Gyroid structure as an example, if G(x, y, z) = 0, then G(x, y, z) < 0 is the physical part, G(x, y, z) > 0 is the pore part. The triple integral of G(x, y, z) = 0 is the volume of a single cell, and the volume fraction V can be expressed as:   

\begin{equation} V = \iiint \frac{\Omega dV}{(X_{\max} - X_{\min}) \cdot (Y_{\max} - Y_{\min}) \cdot (Z_{\max} - Z_{\min})} \end{equation} (3)

  

\begin{equation} \Omega = \left\{(X,Y,Z)\left| \begin{array}{l} \varphi(X,Y,Z) \leq 0,\\ X_{\min} \leq X \leq X_{\max},\\ Y_{\min} \leq X \leq Y_{\max},\\ Z_{\min} \leq X \leq Z_{\max} \end{array} \right|\right\} \end{equation} (4)

Where min to max represents the variation region of Gyroid structure in 3D space, for uniform periodic structure, the range of min to max is 2π. The volume fraction V of a single cell is defined as:   

\begin{equation} \left\{ \begin{array}{l} \rho^{*} = \dfrac{V_{1} - V_{0}}{V_{1}} \times 100\%\\ V = 1 - \rho^{*} \end{array} \right. \end{equation} (5)

Where $\rho^{*}$ is the porosity of a single cell, V₁ is the volume of the outer cube that constitutes the unit, and V₀ is the volume of the unit.

To study the effect of different geometric parameters on the sound absorption characteristics of TPMS sandwich structures, Diamond and Gyroid structures with different volume fractions (V), panel thickness (P), and cell layer number (H) were designed. The design parameters of the two structures are shown in Table 1. The size of the single cell is 5.8 mm × 5.8 mm × 5.8 mm, and there are five cells in the X and Y directions. Taking the DH3C sandwich structure as an example, the design and modeling process is shown in Fig. 1.

Table 1 Design parameters of TPMS sandwich structure.

Fig. 1

Design and molding diagram of DH3C sandwich structure.

2.2 SLM forming Ti6Al4V sandwich structure

Ti6Al4V powder was used in this study. The chemical elements of the powder are shown in Table 2. The less the number of small particles on the surface of the metal powder, the higher the fluidity of the powder will be to a certain extent so that the molded specimen will have a higher forming quality. It can be seen from Fig. 2(a) that the powder surface used in this study is relatively smooth. The powder particle size of SLM process is usually distributed between 10 µm–90 µm, and the powder particle size also affects the density of the forming parts. Figure 2(b) shows that the average particle size of Ti6Al4V powder in this experiment is about 30 µm–50 µm, which is in a narrow distribution range.

Table 2 Mass fractions of different elements in Ti6Al4V powder.

Fig. 2

Particle size distribution of powder.

All specimens in this paper were formed by EPLUS-150 (Beijing e Plus 3D Tech. Co., Ltd.). The main technical parameters are shown in Table 3, and the molded samples are shown in Fig. 3. It can be seen that all sandwich structures have good molding quality without serious adhesion and powder phenomenon.

Table 3 Forming process parameters of Ti6Al4V sandwich structure.

Fig. 3

TPMS sandwich structure formed by SLM.

2.3 Measurement of sound absorption coefficient of TPMS sandwich structure

ISO 10534-2:1998: Determination of sound absorption coefficient and impedance in impedance tubes — Part 2: Transfer-function method²⁴⁾ is used to measure the sound absorption coefficient of TPMS sandwich structure. Bruel & Kjaer impedance tube inner diameter is 29 mm, measuring frequency band is 150–6400 Hz. The test equipment and test principle are shown in Fig. 4.

Fig. 4

Schematic diagram of impedance tube test.

3. Analysis and Conclusion

3.1 Volume fraction

The relationship between volume fraction and sound absorption performance of Gyroid and Diamond sandwich structures is shown in Fig. 5 (panel thickness is 1.5 mm, cell layer number is 3C). The results show that for the Gyroid structure (Fig. 5(a)), the bandwidth of the sound absorption coefficient increases as the volume fraction increases. When the volume fraction is 10%, the first resonance absorption peak appears at 300 Hz and the sound absorption coefficient is 0.13. When the volume fraction is 20%, the first resonance absorption peak appears at 1958 Hz and the sound absorption coefficient is 0.24. The effective absorption frequency range is maintained at about 1500–2500 Hz. As the volume fraction continues to increase, the first resonance absorption peak of GV30 appears at 1977 Hz and the sound absorption coefficient is 0.18, GV40 has no resonant sound absorption over the whole acoustic frequency range. It can be seen that the effect of volume fraction on the sound absorption performance of Gyroid structure increases first and then decreases. Here, a shape factor τ is introduced to describe the bending degree of TPMS cell structure, which is referred to as the tortuous factor for short. The larger the value of τ, the longer the path of the sound wave when it flows through the surface of the structure, and the viscosity loss of the sound wave will increase, to achieve effective sound wave absorption.²⁵⁾ For the TPMS sandwich structure, the volume fraction is small, the internal tortuous factor τ value is low, and the sound absorption performance is not good. As the volume fraction increases, the sound absorption performance increases. When the volume fraction is large and the structure aperture is small, the propagation of sound waves inside the structure is hindered, and less sound energy is converted into heat energy and dissipated, thus reducing the peak value in the sound absorption curve.

Fig. 5

Influence of volume fraction on sound absorption coefficient of TPMS sandwich structure. (a) Gyroid, (b) Diamond.

A similar trend can be observed in Fig. 5(b). Although the structures are different, the volume fraction change has almost the same impact on the sound absorption performance of the two structures. Therefore, the volume fraction in the following study on the impact of panel thickness and cell layer number on the sound absorption performance of TPMS sandwich structure is 20%. In specific applications, TPMS sandwich sound absorption structure has the most appropriate volume fraction value, the volume fraction too high or too low is not conducive to sound absorption, although increasing the volume fraction will improve the sound absorption performance of the structure to a certain extent, for the most complex sound absorption structure, it is difficult to achieve the structure with volume fraction greater than 50%. Because the pore part of this structure will be blocked by the solid part of the material, it is recommended to prefer a light sound-absorbing structure in engineering applications.

3.2 Panel thickness

Figure 6 compares the sound absorption performance of Gyroid and Diamond sandwich structures with different panel thicknesses (volume fraction 20%, cell layer number 3C). The results show that for both Gyroid and Diamond structures, the increase in panel thickness will widen the bandwidth of sound absorption performance. The resonant absorption peaks of GP10 with a panel thickness of 1.0 mm appeared at the acoustic frequency of 745 Hz and 1818 Hz, and the peak values are 0.25 and 0.36, respectively. The resonance absorption peak of GP15 appeared at 1988 Hz, and the peak value is 0.26. GP20 and GP25 have no resonance absorption peak in the whole measurement frequency band. For Diamond structure, DP10 has a resonance absorption peak at 1523 Hz, with a peak value of 0.16. DP15 has a resonance absorption peak at 683 Hz with a peak value of 0.21. It is notable that when the acoustic wave is transmitted from the incident domain to the sample surface in the impedance tube, the panel touches the incident acoustic wave, and the panel plays a role as a hard boundary in the sound absorption problem, and the acoustic wave is almost reflected. Therefore, for the sound absorption structure of TPMS sandwich structure, it is not suitable to design too thick panel thickness. The sound absorption performance of the two structures with panel thickness of 1.5 mm is better, so the panel thickness of 1.5 mm is adopted in the study of the influence law of cell layer number on the sound absorption performance of TPMS sandwich structure.

Fig. 6

Influence of panel thickness on sound absorption coefficient of TPMS sandwich structure (a) Gyroid, (b) Diamond.

3.3 Cell layer number

Figure 7 shows the relationship between panel thickness and sound absorption performance of Gyroid and Diamond sandwich structures (volume fraction 20%, panel thickness 1.5 mm). For the Gyroid structure (Fig. 7(a)), GH2C has the first resonance absorption peak at 894 Hz and the second resonance absorption peak at 2441 Hz, the sound absorption bandwidth is 450 Hz and 478 Hz, and the peak value is 0.32 and 0.31, respectively. GH3C has a resonance absorption peak at 1994 Hz, the absorption bandwidth is 956 Hz, and the peak value is 0.24. GH4C has a resonance absorption peak at 1043 Hz, the absorption bandwidth is 1243 Hz, and the peak value is 0.21. GH5C has the first resonance absorption peak at 1543 Hz and the second resonance absorption peak at 4586 Hz. The sound absorption bandwidth is 1552 Hz and 1852 Hz, and the peak value is 0.23 and 0.19, respectively. It can be seen that the sound absorption bandwidth of Gyroid structure can be widened with the increase of cell layer number. For the Diamond structure (Fig. 7(b)), DH2C has a resonance absorption peak at 2821 Hz acoustic frequency, with a sound absorption bandwidth of 761 Hz and a peak value of 0.16. DH3C has a resonance absorption peak at 589 Hz, the absorption bandwidth is 2050 Hz, and the peak value is 0.20. DH4C has a resonance absorption peak at 1874 Hz, the absorption bandwidth is 1643 Hz, and the peak value is 0.17. DH5C has a resonance absorption peak at 691 Hz acoustic frequency, with a sound absorption bandwidth of 1765 Hz and a peak value of 0.21. With the increase of cell layer number, the overall sound absorption effect of Diamond structure is not good, but the sound absorption bandwidth is high, which first increases and then decreases in the whole range of measured acoustic frequency band. Therefore, when the TPMS sandwich structure is applied to the sound absorption structure, the effective absorption frequency range can be adjusted by adjusting the cell layer number (height) of the sound absorption structure, which should depend on the application requirements in the structure design. The appropriate number of cell layers for sound absorption in the target frequency range should be different for different structure sizes.

Fig. 7

Influence of cell layer number on sound absorption coefficient of TPMS sandwich structure (a) Gyroid, (b) Diamond.

3.4 Structure type

The sound-absorbing performance curves of the Gyroid and Diamond structures are shown in Fig. 8. From the sound absorption coefficient comparison, Gyroid structure is higher than Diamond structure in the range of 150 HZ–6400 Hz, and the sound absorption coefficient of GH2C reaches 0.32 and 0.31 at 894 Hz and 2441 Hz, respectively. The absorption coefficient of GP10 at 745 Hz and 1818 Hz can reach 0.25 and 0.36, respectively. The absorption coefficient of GP15 can reach 0.26 at 1988 Hz. The absorption coefficient of DH5C and DH3C with Diamond structure is about 0.2 at 691 Hz and 589 Hz. The sound absorption bandwidth of Diamond structure is higher than that of Gyroid structure. In terms of sound absorption mechanism, the Gyroid structure mostly exhibits resonance sound absorption mechanism, while the Diamond structure exhibits resonance sound absorption mechanism combined with the viscous loss of sound wave. By comprehensive comparison of the two structures with the best sound absorption performance (Fig. 9). The GP10 sandwich structure with a volume fraction of 20%, panel thickness of 1.0 mm and cell layer number of 3C parameter combination have better sound absorption performance. The sound absorption coefficient is 0.36, and GP10 has a higher sound absorption bandwidth due to the larger aperture of Gyroid structure, which provides a more favorable space for resonant sound absorption. Therefore, for practical engineering applications, suitable structure types can be selected according to specific working conditions for the design of TPMS sandwich sound absorption structure.

Fig. 8

Comparison of sound absorption coefficients between Gyroid and Diamond structures.

Fig. 9

Comparison of sound absorption performance of optimal parameter combination.

4. Discussion

It is well known that there is a relationship between the speed of sound, the wavelength of sound waves, and the frequency at which sound waves are produced:   

\begin{equation} c = \lambda f \end{equation} (6)

Where c is the speed of sound propagation, λ is the wavelength, and f is the frequency.

The experimental environment of this study is room temperature, the measurement frequency band is 200–6400 Hz, and the sound speed is about 343 ms⁻¹. Therefore, the wavelength of sound wave is roughly between 0.054–1.715 m, and the average aperture of TPMS porous structure is much smaller than the wavelength of sound wave. As a result, the surface of the sandwich structure generates very little acoustic scattering.

The acoustic incident energy (IE) is equal to the sum of acoustic reflected energy (RE), absorbed sound energy (AE) and transmitted sound energy (TE).²⁶⁾ Figure 10 shows the transmission process of sound on the surface and core of TPMS sandwich structure. For TPMS sandwich structure, the surface of the panel is rigid and closed (Fig. 10(a)), most of the incident sound waves are reflected, and only a small part of the sound waves may be transmitted through the micro-cracks on the panel surface. The thicker the panel, the smaller the sound energy absorbed through the panel. Therefore, the two TPMS sandwich structures with panel thickness greater than 1.0 mm have poor sound-absorbing ability in the whole measurement frequency band.

Fig. 10

Schematic diagram of sound propagation of TPMS sandwich structure: (a) Surface propagation process, (b) core propagation process.

When a small part of sound waves enters the TPMS sandwich structure (Fig. 10(b)), due to the periodic through channel formed between the core skeleton and the skeleton, most of the sound waves are shot into the through hole, and the remaining sound waves will cause friction with the surrounding air or vibration of the material,²⁷⁾ absorbing and consuming part of the sound energy in the propagation process. Therefore, the Gyroid structure with larger apertures has better sound absorption performance.

5. Conclusion

In this paper, the sound absorption performance of Gyroid and Diamond TPMS sandwich structures was studied by impedance tube method. The effects of volume fraction (V), panel thickness (P), and cell layer number (H) on the sound absorption performance of TPMS sandwich structures were systematically analyzed. The conclusions are as follows:

1. (1)    The effect of volume fraction on the sound absorption performance of Gyroid structure in the range of 10%–40% increases first and then decreases. When the volume fraction is small, the increase of the torsional factor τ increases the sound absorption performance of the two structures, and the volume fraction is too large, which leads to a decrease in the sound absorption performance. Therefore, in the specific application of TPMS sandwich sound absorption structure, there is the most appropriate volume fraction value.
2. (2)    The increase in the thickness of the panel will broaden the bandwidth of the sound absorption coefficient of Gyroid and Diamond structures. The sound absorption performance of the two structures with a thickness of 1.5 mm is better, so it is not suitable to design too thick panels for the Ti6Al4V sandwich sound absorption structure.
3. (3)    With the increase of the cell layer that can broaden the sound absorption bandwidth of Gyroid structure, the sound absorption performance of the two structures in the whole measurement frequency range (150–6400 Hz) increases first and then decreases. In the design of TPMS sandwich, sound absorption structure should depend on the requirements of the application, according to different structure sizes. The appropriate number of cell layers for sound absorption in the target frequency range should vary.
4. (4)    The GP10 sandwich structure with volume fraction of 20%, panel thickness of 1.0 mm and cell layer number of 3C parameter combination has better sound absorption performance and higher sound absorption bandwidth, and the sound absorption coefficient is 0.36.

In conclusion, TPMS sandwich structure has its unique sound absorption ability. It should be noted that TPMS sandwich structure with different macro structure shows different acoustic characteristics, and the structure type is different for different applications. In addition, the TPMS sandwich structure panel in this study plays a role in the sound absorption problem as a hard boundary. Under the condition that the macro structure remains unchanged, the sound absorption performance of the sandwich structure can be greatly improved by a certain number and size of micro-pores on the panel, which is currently discussed from three parts: theory, finite element analysis and experiment.

Acknowledgments

The authors acknowledge the financial support from the National Natural Science Foundation of China (52275278, 51905497), the State Key Laboratory of Mechanical Transmission of Chongqing University (No. SKLMT-MSKFKT-202104), Project funded by China Postdoctoral Science Foundation (2022M721430), and the Graduate Innovation Project of Shanxi Province (2021Y596).

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