# Shunt Damping Vibration Control Technology: A Review
## 3 Passive Shunt
### Single Modal Vibration Control
According to fixed points theory, the analytical optimal natural frequency ratio and the resistance ratio of the RL shunts are determined in terms of displacement, velocity, and acceleration. The optimum tuning parameters in terms of displacement are  
\[f_{Copt} =\left\{ \begin{array}{c}\sqrt{1+\beta}\ \ (\text{series})\\ \sqrt{\dfrac{2-\beta}{2}}\ \ (\text{parallel})\\ \end{array} \right.\] \[\varsigma_{Copt} =\left\{ \begin{array}{c}\dfrac{1}{2}\sqrt{\dfrac{3\beta}{2+ \beta}}\ \ (\text{series})\\ \sqrt{\dfrac{3\beta}{2-\beta}}\ \ (\text{parallel})\end{array} \right. \tag{15}\]  
The optimum tuning parameters in terms of velocity are  
\[f_{Mopt} =\left\{ \begin{array}{c}(1+\beta)\sqrt{\dfrac{2}{2+\beta}}\ \ (\text{series})\\ \dfrac{1\ (\text{parallel})}{4}\\ \dfrac{1}{2}\sqrt{\dfrac{3\beta}{2}}\ \ (\text{parallel})\end{array} \right. \tag{16}\]  
The optimum tuning parameters in terms of acceleration are  
\[f_{Aopt} =\left\{ \begin{array}{c}\dfrac{1+\beta}\ \ (\text{series})\\ \sqrt{\dfrac{2+\beta}{2}}\ \ (\text{parallel})\end{array} \right. \tag{17}\] \[\varsigma_{Aopt} =\left\{ \begin{array}{c}\dfrac{1}{2}\sqrt{\dfrac{3\beta}{1+ \beta}}\ \ (\text{series})\\ \dfrac{1}{2}\sqrt{\dfrac{3\beta}{2}}\ \ (\text{parallel})\end{array} \right.\]  
where \(\beta=\Theta^{2}/KC_{P}\), \(\Theta\) is the modal electromechanical coupling coefficient. The optimized inductance and resistance can be obtained using the optimum \(f_{Aopt}\) and \(\varsigma_{Aopt}\). These three terms of expressions provide a feasible way to optimize RL shunts. Figure 5: An Hard-Disk Drive (HDD) disk-spindle vibration control using piezoelectric bimorph shunt damping, (**a**) schematic diagram, reproduced with permission from [72], IOP Publishing, 2007 and (**b**) photograph, reproduced with permission from [73], IOP Publishing, 2007. In application, piezoelectric elements dissipate some energy due to dielectric loss. This phenomenon is caused by the relaxation time of polarization. [PERSON] et al. [81] considered that dielectric loss can be expressed by a parallel resistance \(R_{C}\) in the equivalent circuit,  
\[R_{C}=\frac{1}{\omega C_{p}\tan\delta} \tag{18}\]  
where \(\delta\) is the dielectric loss factor. As \(R_{C}\) is \(\infty\), there is no energy loss. Similarly, inductors also dissipate energy for they have not only inductance but also internal resistance that can dissipate the energy. Figure 6 shows the improved series and parallel RL circuit considering \(R_{C}\). From (15)-(17), the optimal resistance ratios in the ideal models are almost equal even although these expressions are significantly different. It suggests that the ratio does not rely on the position of the resistance. Hence, the total resistance ratios in the practical models should be tuned to be close to that in the ideal models. The abovementioned series and parallel RL shunts mainly used fixed points theory to find the optimal maximum modal damping. However, the former is suboptimal near resonance due to constructive interference of the two modes with identical frequency, and the latter leads to reduced implemented damping. [PERSON] and [PERSON] [82; 83; 84] proposed an explicit pole placement-based design procedure that relies on equal modal damping and sufficient separation of the complex poles to avoid constructive interference of the two modes. The optimal parameters are [82]  
\[LC\omega_{n}^{2}=\left\{ \begin{array}{c}\frac{1}{\left(1+ \kappa^{2}\right)^{2}}\ \ \text{series}\\ 1\ \text{parallel}\\ \end{array} \right.\] \[RC\omega_{n}^{2}=\left\{ \begin{array}{c}\sqrt{\frac{2 \kappa^{2}}{\left(1+\kappa^{2}\right)^{3}}}\ \ \text{series}\\ \sqrt{\frac{1}{2\kappa^{2}}}\ \ \text{parallel}\end{array} \right. \tag{19}\]  
where \(\kappa^{2}=\theta^{2}/C\omega_{s}^{2}\), \(\theta=v\theta_{p}\), \(v=\mathbf{w}^{T}\mathbf{u}\). \(\theta_{p}\), \(\mathbf{w}\) and \(\mathbf{u}\) are the piezoelectric coupling coefficient, the deformation of the piezoelectric transducer and the modal shape vector.