# Broadening Bandgap Width of Piezoelectric Metamaterial by Introducing Cavity
## 2 Lumped-Parameter Modeling of Piezoelectric Metamaterial
### Semi-Analytical Model of Unit-Cell
After some manipulations (please see Appendix A for details), the governing equations of the unit-cell are obtained as  
\[m\ddot{q}+c\dot{q}+kq+k_{1}(Q_{1}+Q_{2})=0 \tag{11a}\] \[1/C\cdot Q_{1}+k_{1}q =V_{1}\] (11b) \[1/C\cdot Q_{2}+k_{1}q =V_{2} \tag{11c}\]  
where \(m\) is the mass, \(k\) is the stiffness, \(c\) is the damping coefficient, \(q\) is the mechanical displacement, i.e., \(q=\overline{q}e^{-i\omega t}\), \(Q_{1}\) and \(Q_{2}\) are the charges on the transducers on the top and bottom of the substrate, respectively, i.e., \(Q_{1}=(b_{p}l_{p})\overline{D}_{1}e^{-i\omega t}\) and \(Q_{2}=(b_{p}l_{p})\overline{D}_{2}e^{-i\omega t}\), \(C\) is the capacitance of one piezoelectric transducer, \(k_{1}\) is the electro-mechanical parameter between a transducer and the host beam substrate. Specifically,  
\[m=\frac{2\sin(\frac{k_{x}l_{b}}{2})}{k_{x}l_{b}}p_{b}l_{b}b_{b}l_{b}+\frac{2 \sin(\frac{k_{x}l_{b}}{2}\frac{l_{p}}{l_{b}})}{k_{x}l_{b}}2\rho_{p}h_{p}b_{p}l _{b} \tag{12a}\] \[k=2E_{p}l_{b}k_{x}{}^{3}\sin(\frac{k_{x}l_{b}}{2})+4E_{p}l_{p}k_{x}{}^{3}\sin( \frac{k_{x}l_{b}}{2}\frac{l_{p}}{l_{b}})\] (12b) \[k_{1}=-2\frac{E_{p}h_{31}k_{x}}{b_{p}l_{p}}\sin(\frac{k_{x}l_{b}}{2}\frac{l_{p}}{ l_{b}}),\] (12c) \[C=\frac{b_{p}l_{p}}{\beta_{33}h_{p}} \tag{12d}\]  
It is worth noting that, in the unit-cell, two piezoelectric transducers are employed as a bimorph. The top and bottom piezoelectric transducers are identical, and thus have the same capacitance value. The charges, \(Q_{1}\) and \(Q_{2}\), and the voltages across the transducers, \(V_{1}\) and \(V_{2}\), are related to the specific connection configurations, and will be explained subsequently. More importantly, the equivalent mass, stiffness, and electro-mechanical constant are expressed as explicit functions of wavenumber and geometry/material properties. Indeed, in this piezoelectric metamaterial, wave propagates along the continuous beam substrate. The wavenumber indicates the number of waveform cycles per unit length, which decides the corresponding local beam deformation and strain/stress distributions within the unit-cell. Therefore, the equivalent mass, stiffness, and electro-mechanical constant are all wavenumber-dependent. The effect of the circuitry dynamics to unit-cell behavior, where the transducers are connected to inductive shunt in the parallel manner, is now investigated. To derive the governing equations of the integrated system, the electrical dynamics of the unit-cell is examined first. Figure 2 illustrates the equivalent circuit model. The electrodes bracketing the piezoelectric transducers cover fully, the top and the bottom surfaces of each transducer. The electrode layers are connected to an inductive circuit/load. Here the piezoelectric transducers are equated to voltage sources in the equivalent circuit model [28]. A piezoelectric transducer generates electrical displacement due to the mechanical strain, thereby acting as a voltage source. The only cause of mechanical strain is assumed to be the axial strain due to bending wave. The governing equations of a unit-cell integrated with transducers connected with the inductive shunt in the parallel manner, can be written as  
\[w(x,t)=\overline{q}e^{i(k_{x}x-\omega t)}, \tag{10a}\]0 \[w(x,t)=\overline{q}e^{i(k_{x}x-\omega t)}, \tag{10a}\]1where \(L\) is the inductance value adopted under the parallel-connection configuration, and \(R\) is the inherent resistance in the circuitry elements. In this parallel-connection, as the piezoelectric transducers attached to the top and bottom surfaces of the host substrate are identical, the instantaneous charges on the two transducers are equal, i.e., \(Q_{1}(t)=Q_{2}(t)\). To evaluate the unit-cell dynamic characteristics, harmonic responses are assumed with frequency \(\omega\), i.e., \(q(t)=q_{0}e^{i(-\omega t+\varphi_{1})}\), \(Q_{1}(t)=Q_{0}e^{i(-\omega t+\varphi_{2})}\), and \(Q_{2}(t)=Q_{0}e^{i(-\omega t+\varphi_{3})}\), where \(\varphi_{1}\), \(\varphi_{2}\), and \(\varphi_{3}\) are the phases.