# Achieving Enhanced Sound Insulation through Micromembranes-Type Acoustic Metamaterials
## 6 Conclusions  
A theoretical model for calculating the transmission loss through an array of A\(\upmu\)Ms mounted in a lattice frame was presented. To be exact, the model shows how the frame affects the vibration and insulation properties of acoustic metamaterials (A\(\upmu\)Ms) at low frequencies. In designing such an array, estimating the resonant frequency and displacement of the frame is important but difficult. Both thickening and narrowing the frame can reduce its vibration and improve the transmission loss of the entire structure. [PERSON] et al. [14] put forward a lattice-like frame, but the thickness of the frame was 25 mm, which produced a limiting case because the frame was rigid. By comparison, our work is more realistic, using a much thinner frame that is compliant and interacts with the motion of the A\(\upmu\)Ms.  
Our results showed (Section 3) that both the frame and the A\(\upmu\)Ms affect each other's acoustic behavior. When the vibration of the frame is stronger than that of the A\(\upmu\)Ms, increasing the stiffness of the A\(\upmu\)Ms can produce an extremely high TL of the assembly, and vice versa. For the most common case, when the vibration of the frame is much smaller than that of the A\(\upmu\)Ms, increasing the stiffness of the A\(\upmu\)Ms can produce an extremely high STL of the assembly. On the other hand, when the vibration of the frame is strong (corresponding to a super-dense grille-type frame dividing the whole membrane into a large number of ultra-small A\(\upmu\)Ms), increasing the stiffness of the frame will be the first priority. Moreover, if the vibration displacements of the A\(\upmu\)Ms and the frame are about equal, increasing the stiffness of either will have little effect on the TL of the whole assembly. We found that it is important to calculate the vibration amplitude of both the frame and the A\(\upmu\)Ms. Assuming that the aim of the study is to maximize the TL at low frequencies, our approach describes how to match the A\(\upmu\)Ms and the frame so that the target TL can be efficiently achieved.  
In conclusion, A\(\upmu\)Ms have many advantages for low-frequency sound insulation. The governing principle is to ensure that the frame is strong enough to provide rigid boundaries for the A\(\upmu\)Ms, thereby ensuring that the insulation properties are controlled by the stiffness.  
Conceptualization, [PERSON] and X.W.; methodology, J.M.; software, J.M.; validation, J.M., [PERSON] and X.W.; formal analysis, J.M.; investigation, J.M.; resources, [PERSON]; data curation, [PERSON]; writing--original draft preparation, [PERSON]; writing--review and editing, [PERSON]; supervision, [PERSON] All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
**Institutional Review Board Statement:** Not applicable.  
Figure 6: (**a**) The simulated and (**b**) the calculated Sound transmission loss of a 20 \(\times\) 20 cm hole filled with an acrylic plate of 2 mm thick (solid line) and of an A\(\upmu\)M assembly (400 A\(\upmu\)M cells supported by an acrylic frame) (red dashed line). The weights of the two structures are the same (the acrylic frame in the second case is 8.7 mm thick and the A\(\upmu\)M 0.4 mm thick).