# New Optimal Design of Multimode Shunt-Damping Circuits for Enhanced Vibration Control
## 2 Methodology
### Design Optimization
#### 2.4.1 Design Variables  
It is well known that the insertion of an inductance \(L\) in piezoelectric shunt-damping circuits has the effect of cancelling out the piezoelectric transducer impedance \(\mathbf{C}_{p}\) that, in turn, leads to the maximization of the energy dissipation through the resistance \(R\)[28]. This is the role of the inductance \(\widetilde{L}_{i}\) of the shunt branch in the \"current-flowing\" multimode dampener presented in Section 3. However, in cases where low-frequency modes are to be shunt damped, large inductance values are required. This can be a significant constraint because it necessitates the use of impractically large and heavy coil inductors. Several authors [16; 29] proposed the insertion of additional capacitance to lower the values of the required inductances. As stated in [6], the control performance of an \(RL-C\) parallel circuit can be studied by adding an additional capacitance on the shunt circuit, and the values of the inductance and resistance can be reduced. However, the control performance of the shunt circuit may also be reduced by this additional capacitance. Particularly, the control performance of the \(RL-C\) parallel circuit is better when the capacitance value is zero (\(RL\) circuit) and decreases when the capacitance value equalsthe piezoelectric capacitance \(C_{p}\) (\(C=C_{p}\)) and, furthermore, when the capacitance value is two times the piezoelectric capacitance \(C_{p}\) (\(C=2C_{p}\)). [PERSON] et al. [16] suggested inserting an additional capacitance across the terminals of the piezoelectric transducer to drastically reduce the required shunt inductance. For multimode vibration control, a series-parallel impedance structure was proposed with a recommended capacitance value ten to twenty times larger than the piezoelectric capacitance. Although smaller inductances are required in this method, control effectiveness is decreased since additional capacitance allows a decrease in electromechanical coupling. Another method that highlights the role of the capacitance in resonant shunts is the use of negative capacitance to cancel out the transducer capacitance impedance and enhance the vibration attenuation [30, 31, 32]. Generally, the capacitance values \(C_{i}\) of shunt-damping circuits are selected arbitrarily, while the inductances are tuned to a certain frequency to cancel out the transducer capacitance. Particularly in the current-flowing controller introduced in [13, 14, 15], the capacitance values \(C_{i}\) were chosen to be approximately 10% of the piezoelectric capacitance \(C_{p}\), while the capacitance values \(C_{i}\) of the multimode shunt-damping systems in [12, 17] were set to be approximately equal to the piezoelectric capacitance \(C_{p}\). Additionally, in the work of [PERSON] [18], the values of the capacitors were between 1.5 and 3 times the piezoelectric capacitance \(C_{p}\). The arbitrary selection of capacitance values can have a negative impact on the shunt-damping system. The suitable choice of capacitance values is crucial since large capacitance values worsen electromechanical coupling, while small capacitances create a requirement for larger inductance values in the shunt circuit. On the other hand, due to the electrical interaction of the different shunt branches of the multimode shunt-damping circuit, closed-form solutions of resonant shunt branches are subject to significant approximations, resulting in sub-optimal designs. Thus, further fine tuning for electric shunt parameters needs to be performed via an optimization procedure. Taking into account these considerations, instead of arbitrarily choosing the capacitances of the current-flowing circuit and tuning the inductors of each shunting branch into the inherent capacitance of the piezoelectric element, a new optimization approach is proposed in this work that considers them as new design variables along with the resistance values \(R_{i}\).