## 3 Results and Validation
### Validation through Theoretical Transmission Models  
The validation was performed by means of theoretical models and a dedicated experimental setup.  
Two models were considered, both of them focusing on the strain transmission of the piezoelectric device bonded on a flat structural element. Since these models refer to a couple of piezo patches bonded in a collocated configuration and operating in an antagonistic way, an update of the numerical model of the curved beam was performed,  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Adhesive** & **Internal Radii** & **Curvature (1/m)** & **PZT ang.** & **PZT ang.** \\
**Thickness (\(\upmu\)m)** & **(m)** & **Position (deg)** & **Extension (deg)** \\ \hline
80.0 & 0.075 & 13.33 & 45 & 14.75 \\
90.0 & 0.075 & 13.33 & 45 & 14.75 \\
**100.0** & **0.075** & **13.33** & **45** & **14.75** \\
110.0 & 0.075 & 13.33 & 45 & 14.75 \\
120.0 & 0.075 & 13.33 & 45 & 14.75 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Adhesives and properties considered for the parameterization.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Adhesive** & **Shear Modulus** & **Young’s Modulus** & **Poisson’s Ratio** \\  & **(GPa)** & **(GPa)** & **Poisson’s Ratio** \\ \hline Loctite-Hysol-9395 & 1.543 & 4.94 & 0.6 \\
3M Scotch-Weld & 1.5 & 4.5 & 0.5 \\ Epoxy DP490 & 1.461 & 4.237 & 0.45 \\
3M Scotch-Weld & 1.03 & 2.69 & 0.31 \\ Structural AF 3109-2 & 0.342 & 0.958 & 0.4 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Adhesives and properties considered for the parameterization.  
adding also an inner piezo patch. The numerical model presented in Figure 6 was, thus, realized, which reproduces the reference configuration (internal radii and thickness of the beam of 75 and 2.7 mm, respectively; extension and thickness of the piezo patches of 20 and 3 mm, respectively; piezoelectric devices placed at the center of the beam).  
The radial displacement of the free edge was computed by applying an opposite unitary thermal load. In the same analysis, the forces generated by the solving code to simulate the thermal effect were also computed. Both the tip radial displacement and tangential force resultants are reported in Table 7. The forces were equal for the outer and inner piezo patches. In the same table, the radial displacement obtained by replacing the two piezoelectric patches with unitary pin forces is also reported.  
The corresponding transmission factor was computed by dividing the ratio of the radial displacement, i.e., the \(k\) factor defined in (11), by the tangential force generated by the solving code. This parameter, equal to 2.89 \(\times\) 10\({}^{-3}\), was compared in Table 8 with the corresponding ones computed for flat beams with a piezo device of the same dimensions, using the transmission model reported above in Equation (9).  
\begin{table}
\begin{tabular}{c c c c} \hline
**Tip Radial Displacement** & **Tip Radial Displacement** & **Absolute Value of the** & **Absolute Value of the** \\
**Produced by a Unitary** & **Produced by Unitary** & **Tangential Force Applied on** & **Tangential Force Applied on** \\
**Thermal Load (mm)** & **Tangential Pin Forces (mm)** & **the Outer Piezo (N)** & **the Inner Piezo (N)** \\ \hline
3.02 \(\times\) 10\({}^{-8}\) & 6.03 \(\times\) 10\({}^{-7}\) & 17.33 & 17.33 \\ \hline \end{tabular}
\end{table}
Table 7: Outcomes of the analysis of the FE model of the curved beam with two piezoelectric actuators.  
Figure 6: FE model of a curved beam with collocated piezoelectric patches.  
The prediction given by the transmission model on the curved beam was satisfactory. The strain linear profile of the piezo device's thickness, which characterizes this latter model, approximates well the behavior of the piezoelectric device acting on the beam.