# Brake Torque Sensor Calibration Device
## 5 Calculation of the Calibration Device Error
In this calculation, the law of propagation of uncertainties was applied, and the maximum and minimum tolerance values were calculated according to relations (17) and (18):  
\[\sigma_{Mmax}=\sqrt{\sum_{i}M_{i_{max}}^{2}}\quad i=\{2.5;\ 5;\ 10;\ 50\} \tag{17}\]  
\[\sigma_{Mmin}=\sqrt{\sum_{i}M_{i_{min}}^{2}}\quad i=\{2.5;\ 5;\ 10;\ 50\} \tag{18}\]  
where  
\(\sigma_{Mmax/min}\) is the sum of the maximum or minimum torque value induced for a given combination of calibration weights [Nm]. \(M_{i_{max/min}}\) is the maximum or minimum torque value induced by the given weight [Nm]. \[\sigma_{0.5 max} =\sqrt{\sum_{2.5}M_{2.5_{max}}^{2}}=\sqrt{\sum_{2.5}0.50026556^{2}}\] \[r_{max}=\frac{d_{s-U}}{2}+h_{p/2}=\frac{0.4001322}{2}+0.000035=0.2001011\ \text{m}\]0  
All the results of the maximum and minimum torque tolerance values for the given combinations of the calibrated weight are shown in \[r_{max}=\frac{d_{s-U}}{2}+h_{p/2}=\frac{0.4001322}{2}+0.000035=0.2001011\ \text{m}\]1 \begin{table}
\begin{tabular}{c c c c} \hline
**Calibrated Weight** & **Value of Torque \(M\)** & **Min. Value of Torque** & **Max. Value of** \\
**Value [N]** & **[Nm]** & \(M_{i_{max}}\) **[Nm]** & **Torque \(M_{i_{min}}\) **[Nm]** \\ \hline
2.5 & 0.5 & 0.50024294 & 0.50026556 \\
5 & 1 & 1.00049568 & 1.00052131 \\
10 & 2 & 2.00099136 & 2.00104262 \\
50 & 10 & 10.00495686 & 10.00521313 \\ \hline \end{tabular}
\end{table} 
Table 6: Calculated values of maximum and minimum torque values. From these values, the range of changes in determined according was determined according to relation (19):  
\[\pm z_{M_{max}}=\sigma_{Mmax}-\sigma_{Mmin}\quad M=\{0.5;\,1;\,2;\,3;\,4;\,5; \,6;\,7;\,8;\,9;\,10;\,15\} \tag{19}\]  
where  
\(\pm z_{M_{max}}\) is the range of deviations from the nominal value [Nm]. \(\sum M_{max}\) is the sum of maximum torque values for a given combination of calibration weights [Nm]. \(\sum M_{min}\) is the sum of the minimum torque values for the given combination of calibration weights [Nm]. The results of the \(z_{M_{max}}\) calculations are shown in Table 8.  
From the calculated results, the value \(z_{M_{max}}=\frac{\pm z_{M_{max}}}{2}\) necessary for the subsequent calculation of the standard uncertainty of the considered source \(u_{Bz}\) was subsequently determined according to relation (20), as follows:  
\[u_{Bz_{M}}=\frac{z_{Mmax}}{k}\quad M=\{0.5;\,1;\,2;\,3;\,4;\,5;\,6;\,7;\,8;\,9 ;\,10;\,15\} \tag{20}\]  
where  
\(u_{Bz_{M}}\) is the standard uncertainty of the considered source [Nm]. \(z_{Mmax}\) is the range of deviations from the nominal value of torque [Nm]. \(k\) is the value of the chosen approximation of the probability distribution, while for this type of approximation of the distribution, the uniform distribution \(k=\sqrt{3}\) was chosen. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Required Torque** & **Calibrated Weight** & **Combination of Weights** & \
\begin{tabular}{c} **Maximum Torque** \\ **Tolerance \(\sigma_{\text{Mmax}}\) [Nm]** \\ \end{tabular} &
\begin{tabular}{c} **Minimum Torque** \\ **Tolerance \(\sigma_{\text{Mmin}}\) [Nm]** \\ \end{tabular} \\ \hline
0.5 & 2.5 & \(1\times 2.5\)N & 0.50026556 & 0.50024294 \\
1 & 5 & \(1\times 5\)N & 1.00052131 & 1.00049568 \\
2 & 10 & \(1\times 10\)N & 2.00104262 & 2.00099136 \\
3 & 15 & \(1\times 10\)N + \(1\times 5\)N & 2.23723366 & 2.23717635 \\
4 & 20 & \(2\times 10\)N & 2.82990161 & 2.82982912 \\
5 & 25 & \(2\times 10\)N + \(1\times 5\)N & 3.00156393 & 3.00148704 \\
7 & 35 & \(3\times 10\)N + \(1\times 5\)N & 3.60743089 & 3.60733847 \\
9 & 45 & \(4\times 10\)N + \(1\times 5\)N & 4.12525504 & 4.12514937 \\
10 & 50 & \(1\times 5\)N & 10.00521313 & 10.00495686 \\
15 & 75 & \(1\times 50\)N + \(2\times 10\)N + \(1\times 5\)N & 10.44574917 & 10.44548162 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Maximum and minimum torque tolerance values.