## 1 Introduction  
Owing to the band-gap property, periodic structures have attracted a great deal of interest over the past several decades. Periodic beams [1] and stiffened plates [2] are used widely in engineering, while the band gaps are mostly limited to mid-and-high frequency ranges due to the Bragg scattering mechanism [3, 4], which requires the structure period to be of the same order as the wavelength of the band-gap frequency. Since the locally resonant (LR) mechanism was proposed by [PERSON] et al. [5], the study of periodic structures based on the LR mechanism (e.g., phononic crystals [6, 7], acoustic metamaterials [8, 9], etc.) has been increased. They are usually manufactured by adding arrays of resonant microstructures periodically into/onto the structures to suppress flexural wave propagation. It is considered as an efficient method to achieve the low-frequency band gaps. [PERSON] et al. [10] studied the propagation of flexural waves in an LR thin plate made of a two-dimensional periodic array of one type of resonator attached on a thin homogeneous plate. [PERSON] and [PERSON] [11] extended the traditional plane-wave-expansion (PWE) method to calculate the band structures of the \"spring-mass\" simplified model and the \"spring-torsional spring-mass\" simplified model of LR plate systems, where the effect of the visibility is also considered. [PERSON] et al. [12, 13] presented the vibration characteristic of metamaterial beams and plates manufactured out of assemblies of periodic elementswith built-in local resonances. [PERSON] et al. [14] proposed a meta-plate model by periodically attaching high-static-low-dynamic-stiffness resonators onto the thin plate to attenuate very low-frequency flexural waves. To broaden the bandwidths, many researchers investigated the band gaps in periodic structures with multiple local resonators. Flexural wave propagation in locally resonant beams with multiple periodic arrays of attached spring-mass resonators was studied by [PERSON] et al. [15], showing that much broader band gaps can be achieved by applying damped resonators. [PERSON] et al. [16] studied a homogeneous damped plate with parallel attached resonators on both surfaces, from which two band gaps were generated to improve the band-gap performance. [PERSON] et al. [17] investigated wave propagation and power flow in an acoustic metamaterial plate with a lateral local resonance attachment, where two band gaps can be formed by tuning the parameters of the vertical spring-mass and the lateral spring-mass. A chiral-lattice-based elastic metamaterial beam with multiple resonators was numerically and experimentally studied for the broadband vibration suppression by utilizing their individual band gaps by [PERSON] et al. [18]. [PERSON] et al. [19] realized multi-low-frequency band gaps in beams by using the mechanism of multiple resonators containing negative-stiffness. [PERSON] et al. [20] studied the band-gap properties of a periodic vibration suppressor composed of a bottom spring-mass oscillator with multiple secondary oscillators with an arbitrary degree of freedom. [PERSON] and [PERSON] [21] developed the theory of the PWE method to calculate the band structures of the LR plate attached with multiple spring-mass resonators, which is applied in this paper. It was also concluded that the spring damping characteristic can assist in attenuating the vibration responses outside the band-gap frequency ranges. At present, most studies were focused on dynamic behavior based on band-gap properties. It is generally accepted that vibration attenuation leads to noise reduction. Since the band gaps affect the flexural wave propagation, it is expected that the vibro-acoustic radiation from the LR plate can also be suppressed, especially in the low-frequency range. Acoustic radiation from the LR plate-like structures has been investigated by several researchers. [PERSON] et al.