###### Abstract  
In order to suppress the transverse vibration of a plate, a quasi-zero-stiffness (QZS) resonator with tunable ultralow frequency bandgaps was introduced and analyzed. The resonator was designed by introducing the quasi-zero-stiffness systems into mass-in-mass resonators. The plane wave expansion method was employed to derive the bandgap characteristics of the locally resonant (LR) plate with QZS resonators, and corresponding simulations were carried out by finite element method (FEM). The results show that an LR plate with a QZS resonator can provide two bandgaps, and the ranges of the bandgaps agree well with the vibration attenuation bands calculated by FEM. Owing to the introduction of the QZS system, the bandgaps can be easily transferred to a lower frequency or even an ultralow frequency. The damping of the QZS resonators can effectively broaden the vibration attenuation bands. In addition, the differentiated design of the bandgap frequencies can be realized to obtain broadband low-frequency transverse wave suppression performance. Finally, a mechanical structure design scheme was proposed in order to achieve flexible adjustment of the bandgap frequency, which significantly increases the engineering applicability of QZS resonators. ultralow frequency; quasi-zero-stiffness; bandgap; vibration reduction; transverse wave +
Footnote †: journal: Applied Sciences  
0000-0002-2300-3822]B. Xie  
0000-0002-3330-3300-3300-3300-3300-3300-3300-3300-3300-3300-3300-3300-3300-3300-3300-33000-3300-33000-33000-33000-3300-33  
In recent years, the periodic locally resonant (LR) structures have received extensive attention because of their unique bandgap characteristics, which have provided a new idea for the low-frequency vibration control of plates. [PERSON] [10] first clearly proposed the concept of LR structure in 2000 by periodically placing lead blocks wrapped in soft rubber in the polyurethane matrix, which demonstrates that the bandgap frequency is far lower than the Bragg scattering-induced bandgap frequency. Because of its potential for low-frequency vibration control, LR structure has been studied by many scholars since it was proposed [11; 12; 13; 14; 15; 16; 17]. At present, the research on LR structures mainly focuses on broadening and lowering the bandgaps. [PERSON] [18] broadened the vibration attenuation band by combining the Bragg bandgap and the LR bandgap. The numerical simulation results showed that, compared to the plate structures without the LR units, combining periodic LR structures can generate low and wide complete acoustic bandgaps. [PERSON] [19] enlarged the bandgap by constructing the coupling between different modes of the system. The results of the analysis show that a significant enlargement of the relative bandwidth by a factor of three can be obtained through this strategy. [PERSON] [20] designed a lateral local resonator with broad bandgaps by enhancing the coupling among oscillators to suppress the transverse vibration of the plate. [PERSON] [21] designed a metamaterial plate with low and wide frequency bandgaps by using a complex mass-beam resonator to realize broadband vibration suppression. In addition, increasing the resonator's damping is also an effective method to broaden the bandgaps [22]. Compared with enlarging the bandgap width, it is more challenging to lower the frequencies of the bandgaps. For the traditional LR structure, an ultralow frequency bandgap means ultra-large weight or ultra-small static stiffness, which is not allowed in practical applications. Fortunately, the quasi-zero-stiffness (QZS) system [23; 24; 25; 26; 27] used for vibration isolation has the characteristics of high static stiffness and low dynamic stiffness. If the QZS system is introduced into the design of LR structures, the ultralow frequency bandgaps can be obtained without weakening the static stiffness of the structure, and the contradiction between the bandgap frequency and the static stiffness of the traditional LR structure can be overcome.