## 2 Principles of Local and Dual-Connected SSDI Techniques  
### Local Conventional SSDI  
The conventional configuration of SSDI is illustrated in Figure 1a, where \(L\) is the inductor in series with the switch, \(C_{p}\) is the intrinsic capacitance of the piezo-patch, \(V\) is the voltage on the patch. When the circuit is open, the current is zero and so the piezo-voltage varies with the displacement. The switch is closed only when a displacement extreme occurs, where a co-located piezoelectric patch is frequently used to detect the displacement. When it is closed, the capacitance \(C_{p}\) and the inductance \(L\) constitute an electrical oscillator. The switch remains closed until the voltage on the piezo-patch is reversed due to oscillation of the circuit. The closed time is \(\pi\sqrt{LC_{p}}\), namely half a period of the electrical oscillator. The closed time is much less than the open time of the switch because the frequency of the electrical oscillator is much greater than that of structural vibration. In a period of structural vibration, the switch actives twice.  
The relationship between the voltage of SSDI and the structural displacement is illustrated in Figure 2. Theoretically the voltages on piezo-patch before (\(V_{b}\)) and after (\(V_{a}\)) action of the switch are the same. Practically they have difference because of the resistance in circuit, namely:  
\[V_{a}=\gamma_{s}V_{b} \tag{1}\]  
where \(\gamma_{s}\) is the voltage inversion factor:  
\[\gamma_{s}=e^{-\frac{\pi}{\Delta Q_{shunt}}} \tag{2}\]  
and \(Q_{shunt}\) is the electrical quality factor:  
\[Q_{shunt}=\frac{1}{R}\sqrt{\frac{L}{C_{p}}} \tag{3}\]  
From the Figure 2, \(V_{b}\), \(V_{a}\) and the structural displacement also have the following relation:  
\[V_{b}=V_{a}+\frac{2\alpha_{p}}{C_{p}}x_{M} \tag{4}\]  
where \(\alpha_{p}\) is to the force factor, representing the force generated by unit voltage, and \(x_{M}\) represents the displacement extreme of the host structure in each vibration cycle. Generally, \(\alpha_{p}\) is determined by experiment but can also be calculated if the material parameter of the piezoelectric patch is known [31].  
Figure 2: Illustration of the voltage-displacement hysteresis loop of SSDI.  
With Equations (2) and (3), and Figure 2, the voltage on piezo-patch \(V_{\text{shunt}}\big{(}x,\dot{x},t\big{)}\) can be derived [27]:  
\[V_{\text{shunt}}\big{(}x,\dot{x},t\big{)}=\frac{(1+\gamma)\alpha_{p}}{(1-\gamma) C_{p}}\cdot x_{M}\cdot\text{sign}\big{[}\dot{x}(t)\big{]}+\frac{\alpha_{p}}{C_{p}}x(t) \tag{5}\]  
where \(x(t)\) is the displacement of the host structure the force generated by the piezo-voltage in SSDI shunt is proportional to the voltage based on the constitutive equations of piezoelectric material:  
\[f_{\text{shunt}}\big{(}x,\dot{x},t\big{)}=-\alpha_{p}V_{\text{shunt}}\big{(}x,\dot{x},t\big{)}=-\frac{(1+\gamma)\alpha_{p}}{(1-\gamma)C_{p}}\cdot x_{M} \cdot\text{sign}\big{(}\dot{x}(t)\big{)}-\frac{\alpha_{p}}{C_{p}}x(t) \tag{6}\]  
It is clear that in Equation (6) the factor \((1+\gamma)/(1-\gamma)>1\), therefor the absolute value of the first term is always greater than that of second term, which means that the direction of the force \(f_{shunt}\) generated by the piezo-voltage in local SSDI shunt is always opposite to that of velocity \(\dot{x}(t)\). As a result, the force \(f_{shunt}\) plays a role of damping, just like that induced by dry friction.