## 2 Model and Formulations
### Vibration Transmittance of Finite Structures  
In order to observe the bandgap boundaries more clearly, the damping is not considered in the calculation of the above band structures. As is known to all, damping exists in almost all practical equipment, which has a great effect on wave propagation. Hence, it is necessary to study the effect of damping in the QZS resonators on bandgap characteristics. The vibration transmittance of the finite LR plate with varying damping loss factors is shown in Figure 8. The parameter settings in the calculation are the same as those in Table 1. The terms \(\eta_{1}\) and \(\eta_{2}\) represent the damping loss factors of spring \(k_{1}\) and spring \(k_{p}\), respectively. Figure 8: Effects of damping on vibration transmittance of the finite LR plate. Figure 6: Effect of the stiffness ratio \(\beta\) on the \(RBW\) of the first bandgap. Figure 7: Effects of the initial compression ratio \(\gamma\) of the negative stiffness spring on the bandgaps. As can be seen from Figure 8, with \(\eta_{2}\) kept as zero and \(\eta_{1}\) varying from 0 (black solid line) to 0.1 (blue short dot line), the range of the attenuation band near the second bandgap becomes broader while the depth becomes shallower, and the first bandgap changes little. The effect of the damping loss factor \(\eta_{2}\) on the bandgaps is somewhat different from that of the damping loss factor \(\eta_{1}\). As \(\eta_{2}\) increases from 0 (black solid line) to 0.1 (green dash-dot line), the range of the attenuation band near the first bandgap is widened, and the high vibration transmittance near the second bandgap is also effectively suppressed. The reason for this phenomenon is that the effect of \(\eta_{1}\) is only related to the displacement of mass \(m_{1}\), while the effect of \(\eta_{2}\) is related to both displacements of mass \(m_{1}\) and mass \(m_{2}\). Therefore, changing the damping loss factor \(\eta_{2}\) can affect both bandgaps at the same time. Of course, a better bandgap widening performance can be obtained by increasing the factors \(\eta_{1}\) and \(\eta_{2}\) at the same time. In Figure 8, there are still two obvious ultralow-frequency vibration transmittance peaks with frequencies of 7 Hz and 21 Hz, which are undesirable in vibration control. For the traditional resonant structure, it is usually difficult to change its resonance frequency after the structure is finalized. In addition, for low-frequency or even ultralow-frequency vibration control, the mass of the corresponding resonator must be ultra-large, or the static stiffness of the spring must be ultra-small, which is often not allowed in practice. Therefore, it is still difficult for traditional resonators to have both flexible bandgap adjustability and low-frequency bandgap performance. It can be seen from Figure 7 that the bandgaps of the QZS resonator proposed in this paper can be flexibly adjusted to the ultralow frequency range by changing the initial compression of the spring \(k_{n}\) without increasing the mass load. Based on this characteristic, two additional ultralow frequency bandgaps can be generated by adjusting the initial compression of part resonators of the periodic array so as to suppress the two ultralow frequency resonance peaks. In order to adjust parameter \(l_{c}\) conveniently during use, a distance-adjusting device is designed, as shown in Figure 9. Under the premise that the system parameters are unchanged, the initial compression of the negative stiffness spring \(k_{n}\) can be easily changed by rotating the adjusting screw so as to achieve the purpose of adjusting the bandgap frequencies. In order to further obtain two additional low-frequency bandgaps on the basis of Figure 8, the strategy adopted here is to use the structure shown in Figure 9 to differentially design the bandgaps of the QZS LR structure.