## 4 Lower-Frequency Absorptivity Optimization of the Composite Structure  
The frequency of the noise generated by many mechanical devices is concentrated at 800 Hz to 2500 Hz, and the quality and thickness of the sound-absorbing materials are limited by specific environments and conditions. We used OFC3 as the lining material of the composite structure. The overall thickness of the composite structure serves as the basic constraint. The parameters of the composite structure are optimized by the genetic algorithm, in order to improve the sound absorption performance of the composite structure in a specific low frequency band (800-2500 Hz). The integral value of the sound absorption coefficient of the composite structure in a certain frequency band is set to the objective function. The porosity of the perforated plate (\(\phi_{pp}\)), the thickness of the perforated plate (\(L_{pp}\)), the aperture of the perforated plate (\(d_{pp}\)), and the thickness of the intermediate air gap (\(L_{g}\)) are optimized. The objective function can be written as:  
\[obj:max\left[\int_{800}^{2500}\alpha\left(\phi_{pp},\ L_{pp},\ d_{pp},\ L_{g} \right)df\right]. \tag{14}\]  
In practical applications, considering that the composite sound-absorbing structure should satisfy the certain stiffness, the air gap should not exceed 1 mm. Considering the difficulty and cost of processing the perforated plate, the thickness of the PP should not exceed 2.5 mm. The perforation rate constraint of the PP is directly applied in this study. The aperture of the perforated plate determines the processing efficiency and processing cost during processing. Therefore, the minimum aperture is 1 mm. The size of the test piece itself limits the aperture of the perforated plate, and the aperture of the perforated plate is too large, which may result in the loss of characteristics of the perforated plate. The constraints are:  
\[s.t.\left\{ \begin{array}{c}L_{pp}+L_{g}+L_{ps}\leq 20\ \text{mm}\\ 0.01\leq\phi_{pp}\leq 0.2\\ 1\ \text{mm}\leq d_{pp}\leq 3\ \text{mm}\\ 1\ \text{mm}\leq L_{pp}\leq 3\ \text{mm}\\ 0.5\ \text{mm}\leq L_{g}\leq 1\ \text{mm}\\ \end{array} \right.. \tag{15}\]  
The optimized structure parameters are as follows: \(\phi_{pp}\) = 7.26%, \(L_{pp}\) = 2.6, \(d_{pp}\) = 2.6 mm, \(L_{g}\) = 0.99 mm. The optimized PP2 was fabricated using a 3D printer, as shown in Figure 8. The PP2 specimen was accurately and perfectly formed using stereo lithography. The measured structural parameters of PP2 are listed in Table 4.  
Figure 9 shows the measured and calculated results of the sound absorption coefficient of the optimized composite structure. The sound absorption coefficient curve obtained by the semi-empirical model is in good agreement with the measu \begin{table}
\begin{tabular}{c c c c c} \hline
**Sample Number** & **Perforation Ratio, \(\phi_{pp}\) (\%)** & **Hole Diameter, \(d_{pp}\) (mm)** & **Hole Spacing, \(b_{pp}\) (mm)** & **Plate Thickness, \(L_{pp}\) (mm)** & **Specimen Diameter, \(D_{pp}\) (mm)** \\ \hline PP2 & 7.35 & 2.62 & 8.2 & 2.57 & 28.62 \\ \hline \end{tabular}
\end{table} 
Table 4: The measured structural parameters for PP2 test specimen. Figure 8: The optimized composite structure: (**a**) 3D printed PP2 sample; (**b**) Top view; (**c**) Side view. show that the porosity of the perforated plate, the thickness of the perforated plate, the pore size of the perforated plate, and the thickness of the intermediate air layer have significant effects on the sound absorption performance of the entire composite structure. The maximum sound absorption coefficient appears near 1750 Hz for the OFC3 + PP1, and the maximum sound absorption coefficient appears near 1350 Hz for the OFC3 + PP2. The sound absorption peak of the optimized composite structure is shifted to the low frequency direction by 400 Hz, and the OFC3 + PP2 achieves the target of the sound absorption peak in the lower frequency band.