# Measuring Torque and Temperature in a Rotating Shaft Using Commercial SAW Sensors
## 3 Results
### Calibration of SAW Strain Gages
#### 3.3.2 Determination of Gage Factor  
Figure 9a,b show the individual relative resonant frequencies of both gages against the applied strain in the downward and upward directions, respectively, measured at 25 \({}^{\circ}\)C. The gage factors determined with Equation (7) of the individual gages can be seen in Table 4. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Direction of Applied Force**} & \multicolumn{3}{c}{\(S_{G}\)} \\ \cline{2-4}  & **SSE E015** & **SSE E017** & **Differential Output** \\ \hline Downwards & 3.13 & 3.48 & 3.30 \\ Upwards & 3.00 & 4.31 & 3.66 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Values of gage factor of strain gages alone and differential output obtained with force applied in the downward and upward directions. Figure 8: (**a**) Relative resonant frequency of gages SSE15 and SSE17 for different temperature values \(\Delta T\); (**b**) \(\Delta f_{r}\) obtained for different temperature values \(\Delta T\). Figure 10a,b show the difference between the relative resonant frequencies of both gages against the applied strain in the downward and upward directions, respectively, measured at 25 \({}^{\circ}\)C. The gage factor of the load cell (determined with Equation (11)) is \(S_{G}=3.30\) for strain applied in the downward direction and \(S_{G}=3.66\) with strain applied in the opposite direction. The gage factors of both sensors are larger than 3. Since the gage factor of a metallic extensometer is typically 2, systems that use SAW sensors may have enhanced accuracy and precision. However, the gage factor obtained with the different sensors and when the force is applied in opposite directions varies by a factor that can be as high as 1.4. Several factors may originate this effect. On one side, the existence of hysteresis associated with mechanical assembly at the UTM may contribute partially to this difference. On the other side, the asymmetries of the manual gluing process may also explain the slight variation of individual gage factors (for instance when the force is applied in the downward direction, 3.13 against 3.48). However, when the force is applied in the upward direction, the individual gage factors are considerably different (3.00 and 4.31). Even if the eventual existence of hysteresis related with the mechanical system may influence the determination of the gage factor, this large difference can only be explained if the process of gluing the sensors influences their response. While this effect may also take place when strain gages are used, it becomes more problematic with SAW strain gages. An extensometer is composed by a metal wire or film attached to a low Young's modulus carrier film  
Figure 9: Relative resonance frequency of gages SSE E015 and SSE E017 for different values of strain applied in (**a**) downward direction and (**b**) upward direction. Values were measured at 25 \({}^{\circ}\)C.  
like polyimide or kapton. The Young's modulus of the carrier film and the glue used to mount the sensors are similar and any strain or deformation in the aluminum is easily transferred to the gage and the asymmetry in the gluing of the gages has no visible effects. On the other hand, SAW strain gages are fabricated in quartz, which has a Young's modulus similar to that of aluminum (76 GPa for YX and ST cuts [21] against 71.7 GPa for Al7075-T6 aluminum alloy) and, as a consequence, the strain transfer from the aluminum load cell to the piezoelectric material depends on the viscoelastic properties of the adhesive that was used to glue the sensors [22]. Since the gluing of the sensors on the surface of the load cell was made manually, the asymmetries that naturally arise from this process may impact the overall strain transfer and correspondingly the gage factor of the sensors.