# Shunt Damping Vibration Control Technology: A Review
## 3 Passive Shunt
### Multimodal Vibration Control  
In most cases, the single mode shunt is not enough for a multimode system. [PERSON] [28] expanded the single mode shunt damping theory and discussed the effect of the resonant circuit shunted piezoelectrics on a beam, so that a single piezoelectric element can be used to suppress multiple modes. Additional electronic components have been added to the shunt to increase the number of electrical resonances. A capacitor-resistor-inductor shunt is added for each additional structural resonance, as shown in Figure 7a. The electrical impedance of the shunt is  
\[ \begin{array}{l}\overline{Z}_{i}^{EL}(\omega)=\frac{j\omega C_{pi}^{T}}{N}\\ \frac{\sum\limits_{k=0}^{N}\frac{1}{2k}}{2k}\\ Z_{0}=1/(j\omega C_{pi}^{T})\\ Z_{1}=j\omega L_{1}+R_{1}\\ Z_{2}=j\omega L_{2}+R_{2}+1/(j\omega C_{2})\\ \vdots\\ Z_{N}=j\omega L_{N}+R_{N}+1/(j\omega C_{N})\end{array}  \tag{20}\]  
It is difficult to tune the circuit without analytical predictions, since each individual branch of shunt cannot be considered an individual single-mode shunt. Tuning one electrical resonance to suppress a particular mode will detune the rest of the circuit. This shunt is applied to a cantilevered beam and is demonstrated experimentally as a two-mode device. Peak reductions in the second and third bending modes are reduced by 19 dB and 12 dB, respectively. This kind of shunt is called a Hollkamp shunt. However, the optimal inductances for the first mode (\(L_{1}\) = 43.4H, \(L_{2}\) = 33.5H) are too big and make them hard to implement. Another multi-mode shunt that employs a blocking circuit in series with one parallel resistor-inductor shunt circuit is designed to control one modal vibration [75; 87; 88]. The blocking circuit consists of one parallel antiresonant circuit or a series of them. The number of antiresonant circuits determines the modes of structural vibration to be controlled. Figure 7b shows the schematic of two mode shunt damping at \(\omega_{1}\) and \(\omega_{2}\). The first branch, with shunt inductor \(\hat{L}_{1}\) and resistor \(\hat{R}_{1}\), is inserted with a blocking circuit that consists of a parallel capacitor and inductor \(C_{2}-L_{2}\), whose electrical impedance is designed to approach infinity at the second branch frequency \(\omega_{2}\), where \(\omega_{2}^{2}=1/L_{2}C_{2}\). The second branch also utilizes an antiresonant circuit, and \(\omega_{1}^{2}=1/L_{1}C_{1}\). In this way, the remaining branch is effectively open circuit at the resonance frequency of each parallel branch. The total inductance in each branch circuit has to remain the same after the blocking circuit is inserted since the branch circuit has to shunt-damp the corresponding modal vibration. Consequently, the inductance value \(L_{1}^{\prime}\)and \(L_{2}^{\prime}\) of the new shunt has to be adjusted,  
\[ \begin{array}{l}L_{1}^{\prime}=L_{1}-\hat{L}_{2}/(1-\omega_{1}^{2}\hat{L}_{ 2}\hat{C}_{2})\\ L_{2}^{\prime}=L_{2}-\hat{L}_{1}/(1-\omega_{2}^{2}\hat{L}_{1}\hat{C}_{1})\end{array}  \tag{21}\]Figure 7c is the current-flowing shunt, which needs one parallel branch to control each structural mode. The current flowing shunt is similar in nature to the current blocking shunt. Instead of preventing the current flowing at the natural frequency \(\omega_{n}\), the current flowing shunt allows the current to flow [89]. The shaded current flowing network \(\hat{L}_{i}-C_{i}\) in \(i\)th branch is tuned to approximate a short circuit at the target resonance frequency whilst approximating an open circuit at the frequencies of adjacent branches. The remaining inductor and resistor in each branch \(L_{i}-R_{i}\) are tuned to damp the \(i\)th target structural mode in a manner analogous to single-mode shunt damping. Theoretically, the flowing shunt can achieve \(n\) mode vibration control as \(n\) branch circuits connect to the piezoelectric patches.