# A Feasibility Study of a Novel Piezo MEMS Tweezer for Soft Materials Characterization
## 4 Results and Discussion  
To validate the proposed model, numerical simulations have been done. The dimensions and the material characteristics are summarized in Table 1:  
In this work, the FEM results have been chosen as the reference values. Three typical soft tissues (liver, muscle and uterus) of known characteristics [83], have been considered. The value of the beam tip displacement, obtained by the FEM simulations, has been included in the mathematical model to calculate the equivalent Young's modulus and the viscous damping coefficient. The ST sample stiffness can be calculated by means of Equation (8). The values of the Young's modulus and the viscous coefficients reported in the literature have been compared with the ones calculated by the new method. Because of the symmetry of the structure with respect to its mid-plane (see Figure 7) only the upper part has been considered in the simulations.  
In Table 2 the results of the static simulations, obtained by the above described procedure, have been reported. It is possible to observe that the model results are in good agreement with the real values with a percentage error less than 6% in all the cases.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline  & **Material** & **Length** & **Thickness** & **Width** & **Young’s mod. [GPa]** & **Density \([\mathbf{kg/m^{3}}]\)** \\ \hline Beams & Silicon & 750 & 2 & 80 & 170 & 2329 \\ Piezoel. pl. & AlN & 750 & \(2*10^{-2}\) & 80 & 350 & 3300 \\ Clamp teeth & Silicon & 36 & 7.5 & 80 & 170 & 2329 \\ \hline \end{tabular}
\end{table}
Table 1: Beams, piezoelectric plates and end masses specifications; lengths are expressed in \(\upmu\)m.  
To obtain the viscous damping coefficient, dynamics simulations are necessary. As described above, the chosen strategy consists in exciting, by the piezoelectric plates, the \(i\)-th mode of the structure, in order to obtain the amplitude of the free end displacement \(|w_{L}|\) and then in including this in the model.  
A comparison between the eigenfrequencies obtained by the COMSOL FEM code and the proposed model is reported in Table 3.  
In the FEM simulations the first mode has been chosen to excite the structure with the values of the electrical potential reported in Table 4.  
\begin{table}
\begin{tabular}{c c c c} \hline  & \(E\) [kPa] & \(E_{mm}\) [kPa] & \(Err\) \% \\ \hline Liver & 10 & 9.47 & 5.3\% \\ Muscle & 20 & 19.43 & 2.8\% \\ Uterus & 30 & 29.64 & 1.2 \% \\ \hline \end{tabular}
\end{table}
Table 2: Comparison between the actual Young’s modules and those obtained by the model; \(E_{mm}\) is the Young’s modulus obtained by the mathematical model.  
\begin{table}
\begin{tabular}{c c c c} \hline  & \multicolumn{2}{c}{**Liver**} \\ \hline  & \(FEM\) & \(Model\) & \(Err\)\% \\ \hline \(f_{1}[Hz]\) & 15,523.50 & 15,856.40 & 2.14\% \\ \(f_{2}[Hz]\) & 30,701.40 & 29,487.60 & 3.95\% \\ \(f_{3}[Hz]\) & 82,116.70 & 75,668.50 & 7.85\% \\ \(f_{4}[Hz]\) & 162,877.10 & 151,620.00 & 6.91\% \\ \hline \multicolumn{4}{c}{**Muscle**} \\ \hline  & \(FEM\) & \(Model\) & \(Err\)\% \\ \hline \(f_{1}[Hz]\) & 18,768.20 & 18,686.20 & 0.44\% \\ \(f_{2}[Hz]\) & 35,437.60 & 34,681.70 & 2.13\% \\ \(f_{3}[Hz]\) & 82,674.80 & 76,297.90 & 7.71\% \\ \(f_{4}[Hz]\) & 162,995.20 & 151,781 & 6.88\% \\ \hline \multicolumn{4}{c}{**Uterus**} \\ \hline  & \(FEM\) & \(Model\) & \(Err\)\% \\ \hline \(f_{1}[Hz]\) & 20,066.80 & 19,721.70 & 1.72\% \\ \(f_{2}[Hz]\) & 40,190.2 & 39,660.70 & 1.32\% \\ \(f_{3}[Hz]\) & 83,255.70 & 77,038.00 & 7.47\% \\ \(f_{4}[Hz]\) & 163,118.00 & 151,949.00 & 6.85\% \\ \hline \end{tabular}
\end{table}
Table 3: Comparison between the eigenfrequencies obtained by the COMSOL FEM code and the proposed model.  
Figure 7: Scheme used for the simulations.  
Finally, the comparison between the actual viscous damping coefficients and those obtained by the model results have been highlighted in Table 5.  
A good agreement between the relative coefficients is also observed with a percentage error always less than 5%.