## 3 Numerical Simulation Analysis
### Establishment and Validation of a Micro-Perforated Finite Element Model
Considering the overall radiation acoustic power level, in the frequency range of 1-1000 Hz, structures with 20 mm, 30 mm, and 40 mm polyurethane foam have radiation acoustic power levels of 126.76 dB, 126.33 dB, and 126.01 dB, correspondingly. This implies that as the foam thickness increases, the overall radiation acoustic power level decreases. The Influence of Sound-Absorbing Foam Coverage Rate on the Vibration and Acoustic Radiation Performance of Cylindrical Shells  
To investigate the impact of sound-absorbing foam coverage rate on the vibration and acoustic radiation performance of micro-perforated laminated cylindrical shell structures, while keeping the other geometric parameters and material parameters constant, finite element simulation models were established for the influence of 40 mm polyurethane foam within the frequency range. Models with foam coverage rates of 0%, 20%, 60%, and 100% were sequentially created, and the simulated structural radiation acoustic power level frequency spectra are shown in Figure 18.  
Figure 17: Structural radiated sound power level spectra for different foam thicknesses. Figure 18: Structural radiated sound power level spectra for different foam coverage rates. As observed from the graph, in the 1-300 Hz and 810-900 Hz frequency ranges, the structural radiation acoustic power level curves for different foam coverage rates are essentially consistent, indicating that sound-absorbing foam has a lower effect in the low-frequency range and does not significantly affect the structural outward radiation acoustic power. In the 300-810 Hz and 900-1000 Hz frequency ranges, as the foam coverage rate increases, the structural radiation acoustic power level curves noticeably decrease. This is due to the increase in the structure's internal sound energy absorption associated with the increase in polyurethane foam coverage rate, thereby reducing the structural radiation acoustic power level. From an objective standpoint, the structures with 0%, 20%, 40%, 60%, 80%, and 100% coverage of polyurethane foam were tested for overall radiation acoustic power levels within the 1-1000 Hz frequency range. Results showed respective dB levels of 127.2, 126.80, 126.43, 126.26, 126.04, and 126.01. These findings suggest that the micro-perforated laminated cylindrical shell exhibits optimal acoustic performance with full foam coverage. Compared to the scenario without foam, the structures' overall radiation acoustic power levels decrease by 0.4 dB, 0.77 dB, 0.94 dB, 1.16 dB, and 1.19 dB. The reduction magnitude diminishes as the coverage rate increases. The Influence of Sound-Absorbing Foam Placement on the Vibration and Acoustic Radiation Performance of Cylindrical Shells  
The results from Sections 5.3 and 5.4 demonstrate the enhanced vibration and acoustic radiation performance of the structure with varying degrees of thickness and coverage rate of polyurethane foam. Nevertheless, the magnitude of improvement in the structural radiation acoustic power level decreases with an increase in foam coverage rate. To further investigate the impact of foam placement on the vibration and acoustic radiation performance of the structure, this section will examine the effect of the location of the foam placement. To examine the effect of sound-absorbing foam positioning on the vibrational and acoustic radiation performance of micro-perforated laminated cylindrical shell structures, while maintaining constant geometric and material parameters, the foam is divided into five layers based on rib arrangements and numbered from top to bottom as 1-5. Four foam placement methods have been chosen, as demonstrated in Figure 19. All foam placements are polyurethane, and the disparities in the figure are for elucidating the placement methods. Assuming a foam thickness of 20 mm and a coverage rate of 60% inside the cylindrical shell cavity, this section examines the vibration and acoustic radiation performance of the micro-perforated laminated cylindrical shell subjected to different foam placement methods.