## 2 Finite Element Model of Kagome Structure with Piezoelectric Shunt
### Structure with Resonant Shunt Circuits  
A kind of piezoelectric stack transducer is employed (see Figure 4a) to replace a small number of rods in the planar Kagome truss. The transducer is mainly composed of a piezoelectric stack, outer casing, preloading spring and two link rods. A sphere joint is used to prevent the piezoelectric stack from bending and torsion. Each transducer is connected to an \(RL\) shunt (see Figure 4b). The weight of the transducer is about 8 g, which is four times that of the truss rod (2 g). When only a small number of transducers are used, the increment of the total weight can be neglected. The Lagrange-Maxwell equation is employed to build a finite element model of the mechano-electric structure with \(n\) transducers and \(n\) shunt circuits. The kinetic energy of the structure is:  
\[T=\frac{1}{2}\big{\{}\dot{x}_{\mathrm{s}}\big{\}}^{T}[M_{\mathrm{s}}]\big{\{} \dot{x}_{\mathrm{s}}\big{\}} \tag{1}\]  
where \([M_{\mathrm{s}}]\) is the mass matrix of Kagome structure with piezoelectric transducers and \(\{x_{\mathrm{s}}\}\) is the displacement vector of the element nodes. Figure 4: (**a**) Piezoelectric stack transducer; (**b**) Structure with one transducer and one resonant shunt circuit. Figure 3: Mode shapes 1–6 with modal frequencies 297 Hz, 507 Hz, 550 Hz, 717 Hz, 775 Hz, 836 Hz, respectively. The potential energy of the structure is:  
\[V=\frac{1}{2}\{x_{\mathrm{s}}\}^{T}[K_{\mathrm{s}}]\{x_{\mathrm{s}}\} \tag{2}\]  
where \([K_{\mathrm{s}}]\) represents the stiffness matrix of Kagome structure with piezoelectric transducers. The magnetic energy of the inductances is:  
\[W_{\mathrm{m}}=\frac{1}{2}\{\dot{q}\}^{T}[L]\{\dot{q}\} \tag{3}\]  
in which \([L]\) is a diagonal matrix whose elements are value of inductances in the shunt circuits:  
\[[L]=\left[ \begin{array}{cccc}L_{1}&&&\\ &L_{2}&&\\ &&\ddots&\\ &&&L_{n}\end{array} \right] \tag{4}\]  
and vector \(\{q\}\) contains electric charge of different shunt circuits. Supposing all the transducers have the same characters, the electrical energy of the transducers is:  
\[W_{\mathrm{e}}=\frac{1}{2}C\{q\}^{T}\{q\}+\frac{nd_{33}K_{\mathrm{a}}}{C(1-k^ {2})}\{q\}^{T}[B_{\mathrm{t}}]\{x_{\mathrm{s}}\} \tag{5}\]  
where \(C\) is unloaded capacitance of the piezoelectric stack which includes \(n\) piezoelectric disks, \(K_{\mathrm{a}}\) is the short-circuited electrode stiffness of piezoelectric transducer, \(k\) is the electromechanical coupling factor, \(d_{33}\) is the piezoelectric constant and \([B_{\mathrm{t}}]\) is the position matrix of the transducers. The parameters of the stack transducers can be found in [21]. Define the dissipation function \(P\) which represents the energy dissipation of the damping and resistances as:  
\[P=\frac{1}{2}\Big{(}\{\dot{x}_{\mathrm{s}}\}^{T}[D_{\mathrm{s}}]\{\dot{x}_{ \mathrm{s}}\}+\{\dot{q}\}^{T}[R]\{\dot{q}\}\Big{)} \tag{6}\]  
where \([D_{\mathrm{s}}]\) is damping matrix of the mechanical structure and \([R]\) is a diagonal matrix whose elements are value of resistances in the shunt circuits:  
\[[R]=\left[ \begin{array}{cccc}R_{1}&&&\\ &R_{2}&&\\ &&\ddots&\\ &&&R_{n}\end{array} \right] \tag{7}\]  
Combining the mechanical and electrical coordinates as:  
\[\{u\}=\Big{\{}\{x_{s}\}\quad\{q\}\Big{\}}^{T} \tag{8}\]  
the Lagrange-Maxwell equation of the unloaded structure takes the form as:  
\[\frac{d}{dt}\bigg{(}\frac{\partial\mathcal{L}}{\partial\dot{u}_{i}}\bigg{)}- \frac{\partial\mathcal{L}}{\partial u_{i}}+\frac{\partial P}{\partial\dot{u}_ {i}}=0 \tag{9}\]  
where  
\[\mathcal{L}=T-V+W_{\mathrm{m}}-W_{\mathrm{e}} \tag{10}\]Using the Lagrange-Maxwell equation, the dynamical equation of the mechano-electric structure can be obtained as:  
\[[M]\{\dot{u}\}+[D]\{\dot{u}\}+[K]\{u\}=\{0\} \tag{11}\]  
where  
\[[M]=\left[ \begin{array}{cc}[M_{\rm s}]&\\ &[L]\end{array} \right]\quad[D]=\left[ \begin{array}{cc}[D_{\rm s}]&\\ &[R]\end{array} \right]\]  
\[[K]=\left[