## 3 Results  
With the mathematical model and transfer function based analytical model, it is possible to calculate the occurrence of resonances and anti-resonances on the sensory whisker, amid different loading conditions, as shown in Figures 6 and 7. Figure 5: Finite element analysis was performed using COMSOL. Mesh was created with 2907 tetrahedral, 2090 boundary triangular elements, 1048 mesh points and 18,141 degrees of freedom. Finer mesh sizes were tested but no notable difference was seen in the results. Figure 6: Bode plot of the response of the whisker shaft subjected to a point force applied to the end of the beam. Both the outputs have resonances located at the same frequencies due to the same denominators in the transfer functions. At resonance frequencies, the phase decreases by 180\({}^{\circ}\) and is in phase, while at anti-resonance frequencies, the response is 180\({}^{\circ}\) out of phase. The results of finite element modeling of the sensory whiskker shaft have been verified using mathematical and transfer function based models. A small relative error can be seen in Table 2 because the COMSOL based finite element analysis takes into account factors such as bonding between the sensor shaft and the base, changes in geometry as it bends, damping and temperature, which are not accounted for in the analytical model. The COMSOL model also gives the ability to analyse the shape of the whisker at different modes as shown in Figure 8. Using the model, the natural frequency of the whisker can be chosen anywhere between 35 and 110 Hz, by choosing different materials and physical dimensions. The transfer function based model remains important because the COMSOL model does not give anti-resonances, which are given by transfer function based models as shown in Table 3. \begin{table}
\begin{tabular}{l c c c} \hline \hline
**Mode** & **Analytical Model (Hz)** & **FEM 3D (Hz)** & **Relative Error (\%)** \\ \hline
1 st Eigenfrequency & 106.164 & 106.183 & 0.017 \\
2 nd Eigenfrequency & 665.296 & 665.311 & 0.002 \\
3 rd Eigenfrequency & 1862.894 & 1862.271 & 0.033 \\
4 th Eigenfrequency & 3650.555 & 3647.553 & 0.082 \\
5 th Eigenfrequency & 6034.615 & 6026.051 & 0.141 \\
6 th Eigenfrequency & 9014.696 & 8995.007 & 0.218 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: First six modes of vibration calculated using analytical model based on Equation (3) and 3D FEM (Finite Element Model) using the COMSOL package. Figure 7: Bode plot of the response of the whisker shaft subjected to a distributed force. The first (top two panels) case has both resonances and anti-resonances, decreasing and increasing the phase response by \(180^{\circ}\) and \(360^{\circ}\), respectively. The second case (bottom two panels) has only resonances. Sharp edges can be seen as phase changes from \(-180^{\circ}\) to \(180^{\circ}\). With the control system developed for this sensor, the whisking frequency can be varied as desired and vibratory frequency sweeps can be performed between 5 and 236 Hz with acceptable deviation in output as shown in Figures 9 and 10. The minimum standard deviation in sensor output, in terms of frequency is 0.6494 Hz at 200 Hz, while minimum standard deviation in terms of voltage amplitude of sensor output is 0.0012 V at 200 Hz, as shown in Figure 9. The maximum standard deviation in sensor output between 5 and 236 Hz is 6.4127 Hz at 5 Hz, while maximum standard deviation in terms of voltage amplitude of sensor output is 0.0229 V at 5 Hz, as shown in Figure 10.