# Vibro-Acoustic Performance of a Sandwich Plate with Periodically Inserted Resonators
## 3 Results and Discussion
### Acoustic Radiation of a Finite LR Sandwich Plate
In fact, the variation between \(\sigma_{\rm LR}\) and \(\sigma_{\rm HM}\) can be physically explained by the effective mass and how the mass will affect radiation depends on how the energy is radiated. As can be analyzed from the curves of radiation efficiency, zone I is related to monopole radiation and zones II and III correspond to the so-called corner-mode radiation (from 132 to around 2000 Hz) and edge-mode radiation (from around 2000 to 3378 Hz), where sound radiation is mainly limited at the corner or edge of the plate. Figure 11: Comparison of radiation efficiencies between finite and infinite locally resonant sandwich plates. Figure 10: Comparison of radiation efficiencies between a homogeneous sandwich plate and a locally resonant (LR) sandwich plate, together with the normalized effective mass of an LR sandwich plate. Above the coincidence frequency in zone IV, the finite structure behaves as an infinite structure with regards to sound radiation. The normalized effective mass \(m_{e}\) is shown in Figure 10 to further illustrate the effect of \(m_{e}\) on radiation efficiency. In zone I, \(\sigma_{\text{LR}}\) is mainly contributed to from the first several modes, where the first mode makes the greatest contribution, as it has the greatest modal vibration velocity and also the greatest modal radiation efficiency below the fundamental frequency. Thus, both \(\sigma_{\text{HM}}\) and \(\sigma_{\text{LR}}\) can be approximately estimated from the radiation efficiency of the first mode, \(\sigma_{11}=4 abf^{2}/c_{\alpha c}^{2}\), which is the reason it is generally considered as monopole radiation. As shown in this expression, the radiation efficiency is not related to the parameter of mass, and thus the variation in effective mass has little effect on radiation efficiency in zone I. In zone IV, the radiation efficiency of a finite sandwich structure is approximately the same as that of an infinite sandwich structure. Thus, the radiation efficiencies of these two sandwich structures can be expressed as \(\sigma_{\text{LR}}\approx\left(1-f_{\text{clR}}/f\right)^{-1/2}\) and \(\sigma_{\text{HM}}\approx\left(1-f_{\text{chM}}/f\right)^{-1/2}\). As \(m_{e}\) is below unity above 1000 Hz, \(f_{\text{clR}}<f_{\text{chM}}\), which was analyzed in Sub-Section 3.3. Therefore, at most of the frequencies in zone IV, \(\sigma_{\text{LR}}\) is slightly smaller than \(\sigma_{\text{HM}}\). Furthermore, with the increase in frequency, \(m_{e}\) gradually becomes approximately equal to unity, resulting in the radiation efficiency of the LR sandwich plate being nearly the same as that of the homogeneous sandwich plate. In zone II and zone III, for corner-mode and edge-mode radiation, it can be inferred from Equation (25) in [50] that the material density has a negative effect on radiation efficiency. Thus, the increasing mass in zone II induces decreasing radiation efficiency, while the decreasing mass in zone III induces increasing radiation efficiency, especially near the resonance frequency of the resonator. From the above analysis, it can be concluded that the effect of the periodic resonators on radiation efficiency is due to their effect on the effective mass. The effective mass can be used to indicate which part of the frequency region is acoustically improved and which part is acoustically deteriorated. Thus, when the effective mass of an LR sandwich structure is obtained, the radiation efficiency can be estimated to some degree, which gives a rough prediction of the final radiation performance. This can be important for the band-gap design of an LR sandwich plate. The sound radiation power of an LR sandwich plate and a homogeneous sandwich plate are shown in Figure 12, together with the radiation efficiency of the LR sandwich plate.