# A Review on Vibration Control Using Piezoelectric Shunt Circuits
## 4 Modeling of Dynamical Systems with Piezoelectric Shunt Circuits
### Inducing Internal Resonances into the System  
The internal resonance depends on the coupling between two modes whose frequency ratio is an integer. To introduce a 2:1 internal resonance, the modes should have a quadratically nonlinear coupling. Nonlinearity can be induced in active systems by quadratic feedback and control signals [101] in order to activate the saturation phenomenon. In fact, the saturation phenomenon locks one mode's energy and transfers the additional input energy to the other mode, which is extremely helpful in vibration attenuation applications. Also, nonlinearity can be induced in passive and semi-active systems by using nonlinear electrical components such as nonlinear capacitors, in which their capacitance depends inversely on the amplitude of the charge. The implementation of the nonlinear capacitor in the shunt circuit can be represented as follows:  
\[C_{nl}=\frac{1}{Q} \tag{29}\]  
\[V_{nl}=N\frac{Q}{C_{nl}}=NQ^{2} \tag{30}\]  
where Equation (27) becomes  
\[\ddot{Q}+\frac{R}{L}\dot{Q}+\frac{1}{LC_{p}}\left(Q-\sum_{i}^{N}k_{i}q_{i} \right)+NQ^{2}=0 \tag{31}\]  
Figure 8: Frequency response plots of the beam’s tip point without and with a shunt circuit absorber located at the beam’s root, middle, and tip [130].  
By performing the nondimensionalizing and the electro-mechanical modal expansion as demonstrated in [128], the system of equations representing the system in the modal coordinates is given as follows:  
\[\begin{array}{c}\ddot{x}_{1}+\xi_{1}\dot{x}_{1}+\xi_{2}\dot{x}_{2}+\omega_{r}^{ 2}x_{1}+\frac{N}{\sqrt{L^{3}}\,\partial_{1}^{2}}\,x_{1}^{2}+\frac{2N\theta^{2} }{\sqrt{L^{3}}\,\partial_{1}^{2}}\,x_{1}x_{2}+\frac{N\theta^{2}}{\sqrt{L^{3}} \,\partial_{1}^{2}}\,x_{2}^{2}+\frac{C\theta^{4}}{\partial_{1}^{2}}x_{1}^{3}\\ +\frac{3C\theta^{3}}{\omega_{1}^{2}}x_{1}^{2}x_{2}+\frac{3C\theta^{2}}{\omega_ {1}^{2}}x_{1}x_{2}^{2}+\frac{C\theta}{\omega_{1}^{4}}x_{2}^{3}=-\theta\overline {F}\\ \\ \ddot{x}_{2}+\xi_{2}\dot{x}_{1}+\xi_{3}\dot{x}_{2}+x_{2}+\frac{N\theta}{\sqrt{L ^{3}}\,\partial_{1}^{2}}\,x_{1}^{2}+\frac{2N\theta^{2}}{\sqrt{L^{3}}\, \partial_{1}^{2}}\,x_{1}x_{2}+\frac{N\theta^{3}}{\sqrt{L^{3}}\,\partial_{1}^{ 2}}\,x_{2}^{2}+\frac{C\theta^{3}}{\omega_{1}^{2}}x_{1}^{3}\\ +\frac{3C\theta^{2}}{\partial_{1}^{2}}x_{1}^{2}x_{2}+\frac{3C\theta}{\omega_ {1}^{4}}x_{1}x_{2}^{2}+\frac{C}{\omega_{1}^{4}}\,x_{2}^{3}=\overline{F}\end{array} \tag{32}\]  
Numerically simulating the system of equations, [PERSON] et al. [106] found the amplitude-force response and the time history of the host beam and the absorber as shown in Figures 9 and 10, respectively. It is evident that tuning the absorber's frequency to achieve the 2:1 internal resonance is efficient in suppressing the beam's vibration via the saturation phenomenon and transferring the energy to the absorber [106]. Taking the geometric nonlinearity into account, the tuned absorber proves to be effective in suppressing the nonlinear vibration, as shown in Figure 11.  
Figure 10: The time response for the modal coordinates \(x_{1}\) and \(x_{2}\), [106].  
Figure 9: The force–response plots for the modal coordinates \(x_{1}\) and \(x_{2}\), [106].