# Vibro-Acoustic Performance of a Sandwich Plate with Periodically Inserted Resonators
## 3 Results and Discussion
### Acoustic Radiation of a Finite LR Sandwich Plate
The power is expressed in decibels with the reference power selected to be \(6.67\times 10^{-19}\) W. As shown in the figure, in the band-gap (825-1032 Hz), the acoustic radiation of the LR sandwich plate is reduced significantly with an average attenuation of 31.5 dB compared with that of the homogeneous sandwich plate. This improved acoustic performance is mainly caused by the reduction in vibration, as can be seen from Figure 13, yet it should also be noticed that the reduced radiation efficiency in much of the band-gap also makes a positive contribution to reducing sound radiation. It is interesting to observe that the acoustic performance is significantly improved not only in the band-gap, but also in the frequency region close below the band-gap. As can be calculated from 50  
As can be seen in the above results, after periodic resonators are introduced into the sandwich plate, the acoustic performance is improved both in the band-gap and below the band-gap, while the acoustic radiation close above the band-gap is enhanced, owing to both increased radiation efficiency and increased vibration velocity. It is interesting to note that this undesired sound radiation can be suppressed by adding a small damping on the resonator. By doing this, the acoustic performance in the frequency bands above and below the band-gap will both be improved. In order to make the computation concise, hysteretic damping with damping loss factor \(\eta_{R}\) is added to the spring of the resonator, with a complex stiffness expressed as \(\overline{k}_{R}=k_{R}(1+j\eta_{R})\). The sound radiation power of a damped LR sandwich plate with \(\eta_{R}=0.05\) is shown in Figure 14a. It is observed that the damping of the resonator has a minimal effect on sound radiation below 550 Hz and above 1600 Hz. In the band-gap, the maximal attenuation of radiation is decreased when damping is introduced to the resonator. However, at frequencies close below and close above the band-gap, the sound radiation is significantly reduced compared with that of the undamped LR sandwich plate. As shown in Figure 15, with the increase in damping, the sound radiation becomes increasingly smaller in the frequency ranges of 500-900 Hz and 1050-1600 Hz. Figure 12: Comparison of sound radiation power between a locally resonant (LR) sandwich plate and a homogeneous sandwich plate, together with the radiation efficiency of an LR sandwich plate. Figure 13: Comparison of root-mean-square velocity (RMSV) between a locally resonant sandwich plate and a homogeneous sandwich plate. Generally speaking, the damping of the resonator broadens the attenuation zone of sound radiation, especially when the sound radiation above the band-gap is also reduced. In much of the attenuation zone, radiation suppression caused by damping of the resonator is mainly due to the effect of damping on the vibration response instead of the radiation efficiency, which can be inferred by checking Figure 14b,c. For the radiation efficiency in Figure 14c, when \(\eta_{R}\) is set to the small value of 0.05, the damping of the resonator only has an observable effect on the radiation efficiency in the frequency range \(10^{-4}\). Figure 14: (**a**) Sound radiation power, (**b**) root-mean-square velocity (RMSV), and (**c**) radiation efficiency of a homogeneous sandwich plate (HM-SP) and a locally resonant sandwich plate (LR-SP) with damped and undamped periodic resonators. Figure 15: (**a**) Radiation efficiencies and (**b**) radiation power of a locally resonant sandwich plate with various damping parameters for the resonator. range from 745 to 917 Hz, which is located within or below the band-gap. In this frequency band, the damping of the resonator helps to increase the radiation efficiency, which in turn causes increased radiation over 889-917 Hz. As shown in Figure 15a, with the increase in damping, the radiation efficiency is noticeably increased in the frequency band from 745 to 917 Hz, and is decreased near the frequency of 1130 Hz.