# Measuring Torque and Temperature in a Rotating Shaft Using Commercial SAW Sensors
## 3 Results
### Calibration of SAW Strain Gages  
The gages were fixed on the load cell surface using half-bridge configuration with the gages placed at a 45\({}^{\circ}\) angle with the load cell neutral axis in accordance to what is specified in the manual by SENSeOR. With this configuration, the strain felt by each gage \(\varepsilon_{1}=-\varepsilon_{2}\) is given by:  
Figure 7: (**a**) \(\Delta f_{r}\) obtained for different \(\Delta T\). Two dipole antennas were connected directly to the sensor and interrogation unit, respectively. (**b**) Temperature obtained by the thermocouple (black squares) and SAW sensor (red circles).  
\[\varepsilon_{1}=-\varepsilon_{2}=\frac{M\cdot R}{2\cdot J\cdot G}, \tag{6}\]  
where \(M\) is the torque, \(R\) is the radius of the cylinder, \(J\) is the quadratic moment and \(G\) is the shear modulus.  
Similarly to what happens with temperature sensors, the resonant frequency of the SAW gages is given by:  
\[f_{R_{i}}(T,\varepsilon)=f_{R_{i}}(T_{0})\cdot[1+C_{1i}\cdot(T-T_{0})+C_{2i} \cdot(T-T_{0})^{2}+S_{G}\cdot\varepsilon], \tag{7}\]  
where \(f_{R_{i}}(T_{0})\) is the resonant frequency at \(25\)\({}^{\circ}\)C, \(C_{1i}\) and \(C_{2i}\) are the first and second order temperature coefficients, respectively, of each individual resonator, and \(S_{G}\) is the gage factor (\(i=1,2\)) and \(\varepsilon\) is the strain felt by each strain gage. The difference between the relative resonating frequencies of each sensor \(\Delta f_{r}\) is given by:  
\[\Delta f_{r}=\frac{f_{R1}(T)}{f_{R1}(T_{0})}-\frac{f_{R2}(T)}{f_{R2}(T_{0})}= (C_{11}-C_{12})\cdot\Delta T+(C_{21}-C_{22})\cdot\Delta T^{2}+2\cdot S_{G} \cdot\varepsilon. \tag{8}\]  
If the values of the temperature coefficients are the same for both gages, the differential output removes the dependence of the temperature:  
\[\Delta f_{r}=\frac{f_{R1}(T)}{f_{R1}(T_{0})}-\frac{f_{R2}(T)}{f_{R2}(T_{0})}= 2\cdot S_{G}\cdot\varepsilon. \tag{9}\]  
By substituting Equation (6) into Equation (9) and considering that the quadratic moment of the load cell is given by \(J=\frac{\pi\cdot R^{4}}{2}\) (with \(R\) the cell radius), torque at a given temperature \(T\) can be calculated from the difference of the relative resonating frequencies using the following expression:  
\[M=\frac{\pi\cdot R^{3}\cdot G}{2\cdot S_{G}}\cdot\Delta f_{r}. \tag{10}\]  
The calculation of the torque is thus straightforward once the sensors response and gage factor are known. In practice, however, the measurements are not totally independent of temperature. Differences in the process of gluing the sensors on the load cell surface may introduce asymmetries in the coefficients and, as a result, the resonant frequencies depend on temperature:  
\[\Delta f_{r}=C_{2}\cdot\Delta T^{2}+C_{1}\cdot\Delta T+2\cdot S_{G}\cdot \varepsilon, \tag{11}\]  
and Equation (10) becomes:  
\[M=\frac{\pi\cdot R^{3}\cdot G}{2\cdot S_{G}}\cdot\Big{(}\Delta f_{r}-C_{2} \cdot\Delta T^{2}-C_{1}\cdot\Delta T\Big{)}. \tag{12}\]  
As a consequence, the accurate calculation of the torque applied to the cylindrical load cell also requires the knowledge of the coefficients \(C_{1}\) and \(C_{2}\). Therefore, before the strain gage was determined, the response of the gages was measured as a function of the cell temperature.