# A Review on Vibration Control Using Piezoelectric Shunt Circuits
## 4 Modeling of Dynamical Systems with Piezoelectric Shunt Circuits  
### Lumped-Parameter Systems  
The generalized electromechanical equations of motion for multi-degree-of-freedom systems coupled with piezoelectric shunt circuits are developed and presented in the literature by many authors, e.g., [PERSON] et al. [126] and [PERSON] [20] as follows:  
\[[M]\big{\{}\ddot{u}\big{\}}+[C]\big{\{}\dot{u}\big{\}}+[K]\big{\{}u\big{\}}+[K _{c}]\big{\{}v\big{\}}=\{F(t)\} \tag{1}\]  
\[\big{[}C_{p}\big{]}\big{\{}v\big{\}}-[K_{c}]^{t}\big{\{}u\big{\}}=\{Q\} \tag{2}\]  
where \([M]\), \([C]\), and \([K]\) are the system's mass, damping, and stiffness matrices, respectively; \([K_{c}]\) is the electromechanical coupling matrix between the host structure and the piezoelectric material, respectively; \([C_{p}]\) is the inherent piezoelectric capacitance matrix; \(\{F\}\) is the applied mechanical force vector; \(\{Q\}\) is the electric charge vector; \(\{u\}\) is the generalized mechanical coordinates vector, and \(\{v\}\) is the generalized electrical coordinates vector, which is the physical voltage at the piezoelectric electrodes. Moreover, the damping matrix \([C]\) is assumed to be a proportional viscous damping to facilitate modal coupling. For finite element models, Equations (1) and (2) hold with the definition of \([M]\), \([C]\), \([K]\) as the system's global mass, damping, and stiffness matrices, respectively [102; 103; 104; 105; 127]. The piezoelectric capacitance \(C_{p}\) is a function of the geometry and the material of the piezoelectric patch is defined as follows:  
\[C^{\epsilon}=\frac{e^{T}e^{0}l_{p}w_{p}}{t_{p}} \tag{3}\]  
where \(\epsilon^{T}\) and \(\epsilon^{0}\) are the permittivity and piezoelectric charge coefficient. Furthermore, \(l_{p}\), \(w_{p}\), and \(t_{p}\) represent the piezoelectric length, width, and thickness, respectively. The electric voltage across the piezoelectric patch in the Laplace transformed form is as follows:  
\[V_{sh}(s)=-Z_{sh}(s)\ I_{sh}(s) \tag{4}\]  
where \(V_{sh}(s)\) is the voltage across the impedance, \(I_{sh}(s)\) is the current flowing through the impedance, \(Z_{sh}(z)\) is the shunt impedance, and \(s\) is the Laplace operator. The current can also be obtained by differentiating Equation (2) and substituting into Equation (4) to obtain the following:  
\[V_{sh}=-Z_{sh}\Big{(}-[K_{c}]^{t}\{u\}s+[C_{p}]\ V_{sh}s\Big{)} \tag{5}\]  
After rearranging, one can obtain the following:  
\[V_{sh}=\frac{Z_{sh}(s)\ [K_{c}]^{t}\{u\}s}{1+Z_{sh}[C_{p}]s} \tag{6}\]Applying the inverse Laplace transformation for \(V_{sh}\) and substituting the outcome into Equation (1), the following equation is obtained:  
\[[M]\{\ddot{u}\}+\big{(}[C]+Z_{total}[K_{c}]\big{[}K_{c}]^{t}\big{)}\big{\{}\dot{u} \big{\}}+[K]\{u\}=\{F\} \tag{7}\]  
Equation (7) presents the coupled electromechanical equation of motion for a single-degree-of-freedom beam with piezoelectric shunt circuits connected to it. The contribution of the piezoelectric shunt circuit shown in Equation (6) acts as an additional damper for the system. \(\theta\) and \(\theta\) are material properties of the piezoelectric material. The high values of the electromechanical properties of piezoelectric materials allow them to be used as dampers rather than any other material without the need for high inductance values in the circuit. Where the total electrical impedance of the shunt piezoelectric patch, \(Z_{total}\) is given as follows:  
\[Z_{total}=\frac{Z_{sh}}{1+Z_{sh}[C_{p}]\mathrm{s}} \tag{8}\]  
Therefore, the type of electrical elements connected to the piezoelectric patch directly dictates the performance of the shunt circuit. For a circuit that only contains a resistor, the smaller the impedance, the better the performance. This is not the case for all circuit configurations; this will be further discussed in the next section.