# A Review on Vibration Control Using Piezoelectric Shunt Circuits
## 4 Modeling of Dynamical Systems with Piezoelectric Shunt Circuits
### Distributed Parameter Systems
Following the [PERSON]'s method, by substituting Equation (15) into Equation (9), multiplying the resulting equation by the mode shape \(\varphi_{n}\), integrating over the domain from 0 to 1, adding modal damping, and noting the orthogonality conditions and the nondimensional spatial coordinate, we obtain the discretized equations as follows:  
\[\ddot{q}_{n}+2\zeta_{n}\omega_{n}\dot{q}_{n}+\omega_{n}^{2}q_{n}+\sum_{i,j,k}^ {N}a_{ijkn}q_{i}\frac{\partial^{2}}{\partial t^{2}}(q_{i}q_{k})+\sum_{i,j,k}^ {N}b_{ijkn}q_{i}q_{j}q_{k}=f_{n}\cos\Omega t,n=1,2,\ldots,N \tag{16}\]  
where  
\[a_{ijkn}=\frac{1}{2L^{2}}\int_{0}^{1}\left\{\sum_{i}^{N}\varphi_{n}^{\prime}( \overline{x})\int_{1}^{\overline{x}}\left[\int_{0}^{\overline{x}}\sum_{j}^{ N}\sum_{k}^{N}\varphi_{j}^{\prime}(\overline{x})\varphi_{k}^{\prime}(\overline{x})d \overline{x}\right]d\overline{x}\right\}^{\prime}\varphi_{n}(\overline{x})d \overline{x} \tag{17}\]  
\[b_{ijkn}=\frac{EI}{\rho AI^{6}}\int_{0}^{1}\left[\sum_{i}^{N}\varphi_{i}^{ \prime}(\overline{x})\left(\sum_{j}^{N}\sum_{k}^{N}\varphi_{j}^{\prime}( \overline{x})\varphi_{k}^{\prime\prime}(\overline{x})\right)^{\prime}\right] ^{\prime}\varphi_{n}(\overline{x})d\overline{x} \tag{18}\]  
\[f_{n}=\frac{1}{\rho A}\int_{0}^{1}F(\overline{x})\varphi_{n}(\overline{x})d \overline{x} \tag{19}\]  
The first nonlinear term in Equation (16) represents to the inertia nonlinearity and the second term represents the geometric nonlinearity due to the relatively large deformation. The magnitude of the inertia nonlinearity is found to be very small compared with the geometric nonlinearity for the first few modes; hence, it will be neglected in this study. Therefore, discretized equation reduces to the following:  
\[\ddot{q}_{n}+2\zeta_{n}\omega_{n}\dot{q}_{n}+\omega_{n}^{2}q_{n}+\sum_{i,j,k}^ {N}b_{ijkn}q_{i}q_{j}q_{k}=f_{n}\text{cos}(\Omega t),n=1,2,\ldots,N \tag{20}\]  
To mitigate the vibration of the beam by a shunt circuit, the coupling between the beam, piezoelectric patch, and the circuit is considered. Following the formulation of lumped systems attached with shunted circuits by [PERSON] et al. [90] and [PERSON] et al. [104; 105], [PERSON] et al. [106], and [PERSON] et al. [130] developed the coupled equations of the cantilever beam and a passive shunt circuit is derived. The equations representing the beam, piezoelectric patch, and the circuit are as follows:  
\[\ddot{q}_{n}+2\zeta_{n}\omega_{n}\dot{q}_{n}+\omega_{n}^{2}q_{n}+\sum_{i,j,k}^ {N}b_{ijkn}q_{i}q_{j}q_{k}+k_{n}V=f_{n}\text{cos}(\Omega t),n=1,2,\ldots,N \tag{21}\]\[C^{\epsilon}V-Q-\sum\limits_{i}^{N}k_{i}q_{i}=0 \tag{22}\]  
\[L\bar{Q}+R\dot{Q}+V=0 \tag{23}\]  
where \(k_{i}\) is the electromechanical coupling coefficient associated with mode \(i\), \(C^{\epsilon}\) is the capacitance of the piezoelectric patch, and \(V\) and \(Q\) represent the voltage and electrical charges, respectively. The capacitance \(C^{\epsilon}\) is a function of the geometry and the material of the piezoelectric patch defined by Equation (3). \[C^{\epsilon}=\frac{e^{T}e^{0}l_{p}w_{p}}{t_{p}} \tag{24}\]  
The electromechanical coupling coefficient \(e_{i}\) is a mode-shape and material dependent variable defined by the following equation [37]:  
\[\ddot{q}_{n}+2\zeta_{n}\omega_{n}\dot{q}_{n}+\omega_{n}^{2}q_{n}+\sum_{i,j,k}^ {N}a_{ijkn}q_{i}\frac{\partial^{2}}{\partial t^{2}}(q_{i}q_{k})+\sum_{i,j,k}^ {N}b_{ijkn}q_{i}q_{j}q_{k}=f_{n}\cos\Omega t,n=1,2,\ldots,N \tag{16}\]0  
where \(\Delta H=H(\overline{x}-a)-H(\overline{x}-b)\) is the Heaviside function to locate the covered area by the piezoelectric patch. \(E_{p}\) and \(\epsilon^{0}\) are the elastic modulus of the piezoelectric material and the charge coefficient. Moreover, \(L\) and \(R\) are the inductance and resistance values of the \(RL\) shunt circuit. After mathematical manipulation, the voltage in Equation (22) is substituted into Equations (21) and (23) to couple the shunt circuit directly to the structure and optimizing the circuit elements and studying their contribution to vibration suppression.