## 2 Model and Formulations
### Band Structure of Infinite Systems
It can be seen that the theoretically calculated bandgaps coincide with the attenuation frequency bands of vibration response obtained by numerical simulation, which proves the accuracy of the theoretical derivation of the bandgap. In addition, the vibration transmissibility within the bandgap frequency ranges is much smaller than that outside the bandgap, indicating that the propagation of transverse waves is significantly suppressed in the bandgap. In order to more intuitively observe the vibration transmittance of the baseplate inside and outside the bandgaps, Figure 4 shows the displacement response of the baseplate at certain frequencies. It can be seen that, at the frequency within the bandgaps, the vibration is trapped in a small area near the excitation point. However, as the distance from the excitation point to the observation point increases, the vibration response rapidly decays, as shown in Figure 4a,c, which indicates that the bandgaps generated by QZS resonators can effectively suppress the propagation of transverse waves on the plate. For comparison, at the frequencies outside the bandgaps, the wave freely propagates on the plate without attenuation, as shown in Figure 4b,d. The advantage of the QZS system is that the dynamic stiffness of the system can be adjusted arbitrarily by changing the system parameters without changing the static stiffness of the system. It can be seen from Equation (5) that, when the positive stiffness  
Figure 4: The displacement responses of the finite baseplate with QZS resonators at the frequencies within bandgaps: (**a**) 46 Hz and (**c**) 200 Hz; and outside bandgaps: (**b**) 100 Hz and (**d**) 300 Hz. Figure 3: Finite element model of the finite LR plate. \(k_{p}\) remains unchanged, the dynamic equivalent stiffness of the system is only determined by the stiffness ratio \(\beta\) and the negative stiffness spring's initial compression ratio \(\gamma\). Therefore, the effects of terms \(\beta\) and \(\gamma\) on bandgap characteristics are analyzed in detail in the following. The effects of the stiffness ratio \(\beta\) on the bandgaps are shown in Figure 5. It can be seen that both of the two bandgaps move towards a lower frequency range as the term \(\beta\) increases. Especially, the first bandgap is quickly transferred to the ultralow frequency range as the term \(\beta\) approaches 1, which suggests that the QZS LR structure proposed in this paper can obtain a low-frequency or even ultralow-frequency bandgap by adjusting the stiffness ratio of the QZS system, so as to suppress the ultralow-frequency transverse waves. It should be noticed that the first bandgap disappears as \(\beta\geq 1\). The reason is that the value of equivalent dynamic stiffness \(k_{Q}\) becomes negative as \(\beta\geq 1\), which results in mass \(m_{2}\) being unable to resonate. Therefore, the stiffness ratio \(\beta\) of the QZS LR structure needs to be kept below the critical value in a practical application. As \(\beta\) increases, the frequency of the second bandgap shifts towards lower frequencies, and the bandgap width is increased. The ability of the second bandgap to suppress low-frequency vibrations is enhanced, which is a gratifying phenomenon. In order to further assess the structure's band-gap characteristics, the first bandgap's relative band-gap width (\(RBW\)) is defined as  
\[RBW=\frac{\Delta\omega}{\omega_{2}} \tag{19}\]  
where \(\Delta\omega\) is the absolute bandwidth of the first bandgap and \(\omega_{2}\) is the resonant frequency of the mass \(m_{2}\). The effect of the stiffness ratio \(\beta\) on the first bandgap's \(RBW\) is shown in Figure 6. It can be seen that the \(RBW\) decreases slowly with the increase in stiffness ratio \(\beta\), and it still maintains an appreciable width in the ultralow frequency range, which is different from the sharp change of the absolute bandwidth of the first bandgap shown in Figure 5.  
Another important parameter that can adjust the bandgap frequencies is the initial compression ratio \(\gamma\) of the negative stiffness spring.