# Development and Characterization of a Flexible Soundproofing Metapanel for Noise Reduction
## 3 Results
### Structure of Flexible Soundproofing Metapanel
#### 3.1.2 Design of Flexible Soundproofing Metapanel  
Figure 5 provides a comparative analysis of the natural frequencies for unit cell size, offering insight into the dimensional effects on acoustic behavior. Figure 5a shows the STL values as a function of unit cell size. Figure 5b,c depict the 5.5 mm length unit cell exhibiting four distinct natural frequencies, while the 12 mm length unit cell displays numerous natural frequencies, demonstrating how these dimensions affect acoustic radiation characteristics. Specifically, the STL curve for the 12 mm length unit cell shows a multitude of closely spaced dips across the frequency spectrum. This phenomenon can be attributed to the higher number of vibrational modes available in the larger unit cell, each contributing to a localized reduction in STL at its resonant frequency. In contrast, the 5.5 mm length unit cell, with its limited number of natural frequencies within the same range, produces a cleaner STL profile with fewer, more distinct dips. These results illustrate that achieving efficient soundproofing performance in the desired band is challenging, highlighting the importance of selecting appropriate dimensions considering the properties of the thin membrane and frame. Therefore, in this study, we selected a unit cell size of 5.5 mm with fewer dips and a wide bandgap in the target frequency range. In a spring-mass system, the unit cell's resonance can be analyzed as vibrations induced by external forces acting on the thin membrane and the frame. The natural frequency of each mode is determined by its stiffness and area, causing the FSM's STL to vary with material properties and dimensions. The following equation expresses a rectangular plate's flexural rigidity (\(D\)), where \(E\) represents [PERSON]'s modulus, and \(\
u\) denotes [PERSON]'s ratio. This relationship underscores the interconnectedness of material properties and geometric parameters in determining the acoustic performance of the FSM. \[D=\frac{Eh^{3}}{12(1-\mu^{2})} \tag{3}\]  
Consequently, as the natural frequency of each membrane mode is determined by stiffness and area, the STL of the FSM varies with the material properties and dimensions. Figure 6a illustrates the changes in STL due to variations in stiffness, a key factor affecting STL. As the material stiffness increases, the natural frequency values for each membrane mode increase, causing a rightward shift in both the dynamic effective mass and  
Figure 5: Comparison of natural frequencies for different unit cell sizes: (**a**) STL values for various unit cell sizes, (**b**) natural frequencies for 5.5 mm unit cell and 12 mm unit cell, and (**c**) mode shape of the membrane of a unit cell. STL. This relationship implies that structures with high STL values in the desired frequency band can be designed through appropriate material selection. Understanding this relationship is crucial for optimizing the FSM's performance across specific frequency ranges. Rubber materials were chosen for this study to achieve flexibility while maintaining effective sound insulation properties. We compared and analyzed various rubber materials with different moduli, focusing on their acoustic properties and ability to create wide bandgap in the target frequency range. From four candidates ([PERSON]'s moduli: 4.5, 5.5, 6.5, and 7.5 MPa) that exhibited promising characteristics, urethane rubber (6.5 MPa) was selected as the optimal material for the FSM, as shown in Figure 6b. We also analyzed the bandgap position shift in response to membrane thickness variations. As illustrated in Figure 6c, among the four membrane thicknesses examined (0.4, 0.5, 0.6, and 0.7 mm), the 0.6 mm thick membrane demonstrated the most suitable performance within the target frequency range. At this thickness, the bandgap aligns most closely with the target frequency band, and the STL value reaches its peak. Consequently, we selected 0.6 mm as the final membrane thickness.