# Broadening Bandgap Width of Piezoelectric Metamaterial by Introducing Cavity
## 3 Case Analyses and Discussions
### Transmission Analysis  
In this section, transmissions of the piezoelectric metamaterial beam through finite element analysis are presented. Here a piezoelectric metamaterial beam consisting of 12 unit-cells arranged in series is considered (Figure 3b). For simplicity and without loss of generality, the excitation is applied at the left end of the metamaterial beam. The displacement distributions and the forced responses at the right free-end are computed. The dispersion relations are obtained through the displacement distributions of the metamaterial beam by using Fournier transform [25; 39],  
\[W(k_{x})=\frac{1}{2l}\int_{l}w(x)e^{-ik_{x}x}dx \tag{22}\]  
where \(k_{x}\) is the wavenumber and \(w(x)\) is the distribution of displacement in the longitude direction of the beam.  
Plotted in Figure 8, are the transmission diagrams and the dispersion curves of the conventional metamaterial beam and the one with \(25.38\times 12\times 1\) mm\({}^{3}\) cavity incorporated. It can be observed that the bandgaps yielded strong wave attenuation effects in both cases. The piezoelectric metamaterial beam with cavity showed a broader frequency range of wave attenuation. Moreover, the relationship of the bandgap range and the wave attenuation range can be readily observed. Different from that shown by [33], where the frequency range of wave attenuation fully overlapped with the frequency range of bandgap, in this research, significant wave attenuation could be observed outside the bandgap range. For example, wave attenuation occurred between 3195 Hz and 3202 Hz, for the metamaterial beam with cavity. This phenomenon is attributed to the strong resonating (i.e., multiple peaks) near the bandgap, and the corresponding damping from the resonance motion of local resonator.  
Further case studies are carried out to illustrate the influence of the dimension of the cavity to transmissions. Three cavities with the same length (25.38 mm) and height (1 mm), but different widths, 6 mm, 12 mm, and 18 mm, were analyzed. Figure 9 plots correspondingly, three transmission diagrams. It can be readily observed that increasing the width of the cavity can effectively increase the bandgap width. With the cavities, the bandgap widths were 51 Hz, 67 Hz, and 89 Hz, respectively. The transmission diagrams shown in Figure 9, further confirm that incorporating cavity can improve the bandgap behavior.  
Figure 8: Frequency response and dispersion relation. (a) Conventional metamaterial beam; (b) Metamaterial beam with cavity.  
Finally, it is worth noticing that the introduction of cavity does not preclude the optimal arrangement of circuitry elements. For example, tunable negative capacitance elements have been employed in piezoelectric metamaterial synthesis, where the negative capacitance is realized through a negative impedance converter (NIC) [38]. A negative capacitance element can cancel out the inference impedance of the piezoelectric transducer. Equivalently speaking, a negative capacitance element can reduce the system-level stiffness of the unit-cell. This integration can certainly be implanted in combination with the mechanical cavity concept studied in this research. In the past, tunable inductances based on op-amp circuit design have also been employed in vibration control and sensor development [28], as well as in metamaterial synthesis [23]. Integrating tunable inductance with the piezoelectric metamaterial studied in this research, will result in adaptiveness.