## 4 Problem Formulation  
A one-dimensional continuous structure was prepared for the validation of the active control strategy. The structure is a thin, aluminum, cantilevered elastic beam with piezoelectric ceramic patches (PZT) attached to both sides. The specific type of PZT material used is PZT-5A and \"31\" mode has been used for the actuator. It is better for actuator applications for dynamic operating conditions, while it is still good for utilized as sensor applications. The beam can be bent when the ceramic has deformations with applied voltage, which is called a \"bimorph\". Figure 8 shows the concept of different types of bimorph structures. The red arrows show the polarization direction, and green arrows indicate the strain direction. Table 1 shows the material properties of the PZT patch. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Properties** & **Value** & **Properties** & **Value** \\ \hline Thickness & 0.5 mm & Coupling Coefficient \(k_{33}\) & 0.72 \\ Capacitance & 162 nF & Coupling Coefficient \(k_{31}\) & 0.35 \\ Piezo. Strain Coefficient \(d_{33}\) & \(390\times 10^{-12}\) m/V & Polarization Field & \(2\times 10^{6}\) V/m \\ Piezo. Strain Coefficient \(d_{31}\) & \(-190\times 10^{-12}\) m/V & Initial Depolarization Field & \(5\times 10^{5}\) V/m \\ Piezo. Voltage Coefficient \(g_{33}\) & \(24.0\times 10^{-3}\) m/V & Density & 7800 kg/m\({}^{3}\) \\ Piezo. Voltage Coefficient \(g_{31}\) & \(-11.6\times 10^{-3}\) m/V & Curie Temperature & 350 \({}^{\circ}\)C \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Material properties of the piezoelectric ceramic patches (PZT) patch. Figure 8: Bimorph concept: (**a**) parallel bimorph; and (**b**) serial bimorph. Figure 7: Comparison of least mean squares and sliding mode least mean squares: (**a**) change in filter coefficients (Key: w0; w1; w2); and (**b**) comparison of trajectories in the plane of filter coefficients (Key: least mean squares; sliding mode least mean squares). Figure 9 shows a schematic of the experimental setup. Serial bimorph PZT actuators are attached to the surface of the aluminum beam, and an additional small piezoelectric ceramic is attached to one side as a strain sensor for the feedback control system. PZT sensors are selected to check their feasibility for the active vibration control applications. An accelerometer is attached at the tip of the beam as a performance indicator. A bench vise and an electrodynamic shaker are installed on a vibration isolation table, and the whole beam structure is gripped with the vise, resulting clamped-free boundary conditions. The electrodynamic shaker provides impulsive force to the vise, which results in lateral vibration of the beam that has to be mitigated. Additional mass can be attached to the right side of the beam, which works as an impedance change of the structure. A data acquisition and control system (dSPACE 1104) is used to implement real-time control, and it is connected to a desktop computer with MATLAB/Simulink installed. The designed controllers are uploaded to the dSPACE module after compiling the MATLAB files, and the control system is prepared as follows: (1) signals from the PZT sensor and the accelerometer are acquired by ADC channels and used for the control; and (2) the control input calculated in the dSPACE module is produced from a DAC channel and provided to the PZT actuators after amplification. Figure 10 shows the main components of the experimental setup. The following frequency response function was obtained from the impact test, and the relationship between the input and output (the transfer function) can be predicted from the results. The objective is to investigate the controller performance with this baseline structure and a modified structure with additional mass, which dramatically changes the structural dynamics and transfer function. Figure 11 compares the frequency response functions of the baseline and the modified structures, and obvious changes in natural frequencies can be observed. Figure 10: Experimental setup.