# A Review on Vibration Control Using Piezoelectric Shunt Circuits
## 4 Modeling of Dynamical Systems with Piezoelectric Shunt Circuits
### Distributed Parameter Systems  
In this section, we develop a mathematical model for a continuous system connected to a piezoelectric shunt circuit. The model describes the multi-modal electromechanical behavior of the system. The approach presented here can be extended to one-dimensional and two-dimensional structures such as beams and plates with general boundary conditions. The model accounts for the geometric and inertia nonlinearities arising from the characteristics of the structure. Without loss of generality, we consider an Euler Bernoulli cantilever beam with an RL shunt circuit in series. The nonlinear planar vibration of a cantilever beam is given by [PERSON] et al. [129] as follows:  
\[\rho A\frac{\partial^{2}w}{\partial t^{2}}+EI\frac{\partial^{4}w}{\partial x^{ 4}}+\frac{\rho A}{2}\bigg{\{}\ w^{\prime}\int_{L}^{x}\frac{\partial^{2}}{ \partial t^{2}}\ \Big{[}\int_{0}^{x}\left(w^{\prime}\right)^{2}dx\Big{]}dx\bigg{\}}^{ \prime}+EI\Big{[}w^{\prime}\big{(}w^{\prime}w^{\prime\prime}\big{)}\Big{]}^{ \prime}=F\text{sin}\,\Omega t \tag{9}\]  
where \(w\) is the displacement of the beam; \(\rho,E,A\) and \(I\) are the density of the beam, modulus of elasticity, cross-sectional area, and second moment of inertia of the beam, respectively; \(F\) is the amplitude of the applied harmonic force and \(\Omega\) is its frequency; and \(x\) is the spatial coordinate along the length of the beam and \(t\) is time. The first nonlinear term in Equation (9) is the inertia nonlinearity, and the second term is the geometric nonlinearity. Introducing a nondimensional spatial coordinate \(\overline{x}\) such that \(\overline{x}=x/l\) and solving the linear-free vibration problem yields the following mode shapes and natural frequencies:  
\[\varphi(\overline{x})=\text{sin}(V_{i}\overline{x})-\text{sinh}(V_{i}\overline {x})-\frac{\text{sinh}(V_{i})+\text{sin}(V_{i})}{\text{cosh}(V_{i})+\text{ cos}(V_{i})}(\text{cos}(V_{i}\overline{x})-\text{cosh}(V_{i}\overline{x})) \tag{10}\]  
\[\omega_{i}=\sqrt{\frac{EI}{\rho AI^{4}}}V_{i}^{2} \tag{11}\]  
where the constants \(V_{i}\) are frequency coefficients governed by the following characteristic equation:  
\[\text{cos}\,V_{i}\text{cosh}\,V_{i}+1=0 \tag{12}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Circuit** & **Circuit Configuration** & **Total Impedance** & **Characteristics** & **References** \\ \hline \hline \end{tabular}
\end{table} 
Table 1: _Cont._Moreover, the mode shapes satisfy the following orthogonality conditions:  
\[\int_{0}^{1}\varphi_{i}(\overline{x})\varphi_{j}(\overline{x})dx= \begin{cases}1, \ i=j\\ 0,\ i\
eq j\end{cases}  \tag{13}\]  
\[\int_{0}^{1}\varphi_{i}(\overline{x})^{\prime\prime\prime\prime}\varphi_{j}( \overline{x})dx= \begin{cases}V_{i}^{4},\ i=j\\ 0,\ i\
eq j\end{cases}  \tag{14}\]  
The [PERSON]'s method is used to discretize the governing partial differential equation, Equation (9), into a set of ordinary differential equations. Therefore, we let the displacement \(v(\overline{x},t)\) be discretized as follows:  
\[w(\overline{x},t)=\sum_{i=1}^{N}\varphi_{i}(\overline{x})q_{i}(t) \tag{15}\]  
where \(\varphi_{i}\) are the linear vibration mode shapes, \(q_{i}\) are generalized coordinates to be determined, and \(N\) is the number of modes retained in the discretization.