# New Optimal Design of Multimode Shunt-Damping Circuits for Enhanced Vibration Control
## 2 Methodology
### State-Space Model of the Structure with Current-Flowing Circuit  
In this section, we derive the state-space representation of the compound system in Figure 1 when a \"current-flowing\" shunt circuit is connected to the top piezoelectric patch. The derivation is presented for the case of two modes of vibration control at the target resonance frequencies, i.e., \(\omega_{1}\) and \(\omega_{2}\) (\(\omega_{1}<\omega_{2}\)). The extension to multiple-mode control is straightforward. The simplified current-flowing controller for the two-mode case is shown in Figure 3.  
Figure 3a shows the equivalent circuit design of the shunt attached to a piezoelectric element where capacitor \(C_{p}\), in series with a strain-dependent voltage source \(V_{p}\), represents the top piezoelectric element. Next, we use the symbol \(q\) to denote the electric charge of the top piezoelectric patch (i.e., \(q=Q_{1}\) in Equation (31)). Moreover, Equation (32) is automatically met and may be disregarded in the scenario when the bottom piezoelectric patch serves as the actuator. The corresponding term in Equation (30) can, thus, be shifted to the right-hand side as an equivalent electrical work:  
\[M_{u}\ddot{\vec{d}}+C_{i}\dot{\vec{d}}+K_{u}\vec{d}+\mathbf{\Theta}_{1}v_{1}=F_{m }-\mathbf{\Theta}_{2}v_{2} \tag{39}\]  
Figure 2: Current-flowing shunt circuit for multimode damping. To derive the electrical circuit equations, [PERSON]'s circuit laws are applied to the simplified shunt circuit of Figure 3b:  
\[\dot{q}=\dot{q}_{1}+\dot{q}_{2} \tag{40}\]  
\[L_{1}\ddot{q}_{1}+R_{1}\dot{q}_{1}+\frac{1}{C_{1}}q_{1}=-V_{sh} \tag{41}\]  
\[L_{2}\ddot{q}_{2}+R_{2}\dot{q}_{2}+\frac{1}{C_{2}}q_{2}=-V_{sh} \tag{42}\]  
We know that the piezoelectric voltage and shunt voltages must be equal (\(v_{1}=V_{sh}\)) and that the current from the piezoelectric element matches the current delivered to the shunt circuit. Solving Equation (31) with respect to \(v_{1}\), substituting into Equation (39) and using Equations (40)-(42), we obtain the final electromechanical equations of motion for a beam and piezoelectric element connected to the shunt circuit of Figure 3b:  
\[\ddot{\mathbf{d}}=-M_{u}^{-1}\mathbf{K_{o}}\mathbf{d}-M_{u}^{-1}\mathbf{\Theta}C_{p}^{-1}q+\mathbf{ M}_{u}^{-1}\mathbf{F}_{m}-\mathbf{M}_{u}^{-1}\mathbf{\[PERSON]}_{2}v_{2} \tag{43}\]  
\[\dot{q}=\dot{q}_{1}+\dot{q}_{2} \tag{44}\]  
\[\ddot{q}_{1}=-\frac{1}{L_{1}C_{p}}\mathbf{\[PERSON]}_{1}^{T}\mathbf{d}-\frac{1}{L_{1}C_{ p}}q-\frac{1}{L_{1}C_{1}}q_{1}-\frac{R_{1}}{L_{1}}\dot{q}_{1} \tag{45}\]  
\[\ddot{q}_{2}=-\frac{1}{L_{2}C_{p}}\mathbf{\[PERSON]}_{2}^{T}\mathbf{d}-\frac{1}{L_{2}C_{ p}}q-\frac{1}{L_{2}C_{2}}q_{2}-\frac{R_{2}}{L_{2}}\dot{q}_{2} \tag{46}\]  
where  
\[M_{u}\ddot{\vec{d}}+C_{i}\dot{\vec{d}}+K_{u}\vec{d}+\mathbf{\Theta}_{1}v_{1}=F_{m }-\mathbf{\Theta}_{2}v_{2} \tag{39}\]0  
By defining the state vector as \(\mathbf{x}_{sh}=\left\{\mathbf{d},\dot{\mathbf{d}},\,q,q_{1},q_{2},\dot{q},\dot{q}_{1}, \dot{q}_{2}\right\}^{T}\), Equations (43)-(46) can be combined and written as:  
\[M_{u}\ddot{\vec{d}}+C_{i}\dot{\vec{d}}+K_{u}\vec{d}+\mathbf{\Theta}_{1}v_{1}=F_{m }-\mathbf{\Theta}_{2}v_{2} \tag{39}\]1  
\[M_{u}\ddot{\vec{d}}+C_{i}\dot{\vec{d}}+K_{u}\vec{d}+\mathbf{\Theta}_{1}v_{1}=F_{m }-\mathbf{\Theta}_{2}v_{2} \tag{39}\]2 (a) A single-shunt circuit for two-mode current-flowing control. (b) Simplified shunt circuit. The state matrices are given by:  
\[A^{sh}=