# Shunt Damping Vibration Control Technology: A Review
## 3 Passive Shunt
### Multimodal Vibration Control
The inductance of the shunt branches is  
\[L_{1}=\frac{1}{\omega_{1}^{2}C_{p}},L_{2}=\frac{1}{\omega_{2}^{2}C_{p}}, \cdots,L_{N}=\frac{1}{\omega_{n}^{2}C_{p}} \tag{22}\]  
If the capacitance of the current flowing branches is \(C_{1},C_{2},\cdots,C_{n}\), the inductance of the current flowing branches is  
\[\hat{L}_{1}=\frac{1}{\omega_{1}^{2}C_{1}},\hat{L}_{2}=\frac{1}{\omega_{2}^{2} C_{2}},\cdots,\hat{L}_{N}=\frac{1}{\omega_{n}^{2}C_{n}} \tag{23}\]  
The current-flowing shunt can be modified and simplified by combining values of series inductor together, e.g., \(L_{sumi}=L_{i}+\hat{L}_{i}\)[90]. The current-flowing network decouples the multi-mode problem into a number of approximately independent single-mode problems. Unlike the current-blocking technique, the order of each current-flowing branch does not increase, with the number of modes to be controlled simultaneously increasing. Moreover, the current-flowing shunt requires less components and gracefully extends to damp a large number of modes simultaneously. The vibration control performance of a plate shows that the current-flowing shunt can suppress the first five resonant amplitudes by 3.8, 10.1, 12.8, 13.2, and 14.7 dB, respectively. The corresponding optimal inductance of the first five mode vibration are 1986.3H, 491.1H, 259.2H, 142.2H, and 71.1H, respectively. [PERSON] and [PERSON] [91] considered a controlled source that establishes the relationship between the controlled current and voltage. The impedance of the controlled source can be regarded as the equivalent impedance of the shunt circuit. Hence, an arbitrary impedance can be realized. [PERSON] and [PERSON] [34] extended the current flowing shunt damping from the piezo to the electromagnetics. An electromagnetic coil was bonded on the beam and a fixed permanent magnet was placed under the bottom of the coil. The terminals of the coil connected to the current-flowing shunt circuit to control two modal vibrations. Similar to the piezoelectric shunt, the current-flowing shunted  
Figure 7: Multimode shunts, (**a**) Hollkamp shunt; (**b**) current-blocking shunt; (**c**) current-flowing shunt; and (**d**) series-parallel shunt. electromagnetic of a single mode branch is the series of capacitor-inductor-resistor. The multiple mode vibration can be controlled by adding parallel shunts. The optimal frequency \(\gamma_{opt}\) and the optimal inductance of the shunt \(L_{opt}\) were obtained according to fixed points theory, which are  
\[ \begin{split}\gamma_{opt}=\sqrt{\frac{2-\psi}{2}}\\ L_{opt}=\frac{2M_{i}+C_{1}\phi^{2}}{2K_{i}C_{1}}\end{split}  \tag{24}\]  
where \(\psi=\phi^{2}/M_{i}(L+L_{1})\omega_{i}^{2}\) is the electromagnetic coupling coefficient. \(M_{i}\), \(K_{i}\) and \(\phi\) are the modal mass, stiffness and shape vector of the system, respectively. \(L_{1}\) is the inductance of the shunt for mode one. The experiment results showed that the first and second mode amplitudes were reduced by 25 and 10 dB. The passive multimode shunts mainly use electrical resonant characteristics to produce the multimode vibration control performance, e.g., current-flowing, current-blocking. Figure 7d presents another multimode shunt using a series-parallel circuit, which can reduce inductance of the shunt [92]. Each network in series \(\hat{L}_{i}-\hat{C}_{i}-L_{i}-R_{i}\) contains two sub-networks: a current-blocking network \(\hat{L}_{i}-\hat{C}_{i}\), and a parallel single-mode damping network \(L_{i}-R_{i}\). The two networks are tuned to the same target resonance frequency \(\omega_{i}\). Under this frequency, the current-blocking network has an extremely large impedance. All of the remaining current-blocking networks, tuned to other structural resonance frequencies, have a low impedance at \(\omega_{i}\). The voltage applied at the terminals results in a current that flows freely through the detuned current-blocking networks but is forced to flow through the active damping network.