# Shunt Damping Vibration Control Technology: A Review
## 4 Active Shunt
An improved circuit has been constructed shown in Figure 10. The inductance ratio \(\beta_{m}\) is given as  
\[\beta_{m}=-(R_{1}/R_{2})[L_{s}(1-\text{itan}\delta)/L_{e}(1-\text{itan}\delta)] \tag{29}\]  
Under dynamical vibration, the actuator should have a loss factor and the inductance can be expressed by the complex quantity \(L_{s}(1-\text{itan}\delta)\), where \(\tan\delta=R_{0}/\omega L_{0}\). It is observed that the resonant frequency of the system decreases or increases, depending on the type of negative inductance shunt. After that, [PERSON] et al. [PERSON] et al. (2016) used a negative inductance shunt as an electrically-controlled element to develop a variable-stiffness spring based on magnetostrictive transducers. There is an alternative active shunt called the active elasticity control (AEC) method proposed by [PERSON] et al. [PERSON] et al. (2016). It is another name for the active shunt, but it has special design analysis procedures. The AEC is realized by connecting the piezoelectric element to an external feedback circuit. The induced voltage of piezoelectric transducers generates secondary deformation due to the inverse effect. Because  
Figure 10: Improved negative inductance circuit. the induced voltage depends on the internal and external impedances, the change of the external impedance affects the additional deformation of the transducer due to the inverse effect. Then the total deformation can be controlled by controlling the impedance of the external shunt. The parameter that controls the effective stiffness of the piezoelectric transducer is the capacitance. If the external shunt circuit with impedance \(Z_{\text{ext}}\) connects to a piezoelectric transducer, the ratio of the impedance of the inherent capacitance \(C_{p}\) and \(Z_{\text{ext}}\) is \(\alpha=1/(i\omega C_{p}Z_{\text{ext}})\) [PERSON] et al. [109] suggested that the elastic compliance \(s(\alpha)\) can be deduced according to (1),  
\[s(\alpha)=s^{E}[1-k^{2}/(1+\alpha)] \tag{30}\]  
where \(k^{2}=d^{2}/(s^{E}\zeta^{T})\). If a negative capacitance shown in Figure 9 is used, the dynamic elastic constant of piezoelectric transducers will be modified. When the negative capacitance shunt was used, the elastic constant increased significantly, and decreased when the equivalent capacitance of the circuit was equal to that of the transducer. When an inductance shunt was used, the elastic constant exhibits resonance and anti-resonance against frequency around the electrical resonant frequency of the circuit. The elastic loss was significantly modified by changing the compensatory resistance. The AEC method that controlled the elastic property of piezoelectric transducers has become the basis of smart structural vibration control, and it provides a possibility to design new vibration and noise control devices. [PERSON] et al. [110,111] carried out the theoretical analysis of AEC devices, and the result of negative capacitance shunted AEC showed that a 35 dB transmission loss reduction in the frequency range from 100 Hz to 2 kHz was achieved. After that, various applications of AEC were demonstrated by [PERSON] et al. [112]. [PERSON] et al. [113] designed a vibration isolation system utilizing AEC to control the effective elastic stiffness of piezoelectric transducer. An additional feedback control was introduced to realize the automatic stabilization and adjustment of the negative capacitance. Based on this work, [PERSON] et al. [114] proposed an adaptive negative capacitance shunt to enhance operational stability of a sound-shielding system. [PERSON] and [PERSON] [115] established the effective anisotropic Young's modulus of negative capacitance using finite element modeling. The AEC method provides an alternative approach to the design shunt damping devices and can be very efficient in solving complicated vibration and noise problems.