## 3 Results and Discussion  
### Calculation of Thrust Measurement Uncertainty  
According to the methodology applied in the manuscript [13] and with the purpose of gathering data to perform a statistical analysis, 10 compression tests were conducted, applying a force of 2500 N, using a standard tensile and compression machine. Table 2 describes the ten values indicated by the acquisition system and Type A uncertainty of results. Applying the partial derivative of Equation (7) to the mathematical model of the acquisition system represented by Equation (5), and using the mathematical transformations based on Equations (2)-(4), the following sensitivity coefficients are obtained:  
\[\text{Coefficient}\ V_{out\ eff}\text{: }\frac{\partial\left(\frac{V_{out\ eff }-V_{0}}{C_{f}}\right)}{\partial V_{out\ eff}}=\frac{F}{V_{out\ eff}-V_{0}} \tag{10}\]  
\[\text{Coefficient}\ C_{f}\text{: }\frac{\partial\left(\frac{V_{out\ eff }-V_{0}}{C_{f}}\right)}{\partial C_{f}}=\frac{-F}{C_{f}} \tag{11}\]  
\[\text{Coefficient}\ V_{0}\text{:}\frac{\partial\left(\frac{V_{out\ eff}-V_{0}}{C_{f}} \right)}{\partial V_{0}}=\frac{-F}{V_{out\ eff}-V_{0}} \tag{12}\]  
Table 3 lists the sources of uncertainty involved in determining the thrust measurement. \begin{table}
\begin{tabular}{c c c c} \hline
**Test Number** & **Result (N)** & **Mean (N)** & **Standard Deviation (N)** \\ \hline
1 & 2497.24 & & \\
2 & 2497.77 & & \\
3 & 2498.40 & & \\
4 & 2498.92 & & \\
5 & 2498.35 & & \\
6 & 2499.29 & & \\
7 & 2498.20 & & \\
8 & 2498.94 & & \\
9 & 2497.74 & & \\
10 & 2499.26 & & \\ \hline \end{tabular}
\end{table} 
Table 2: Measurement results. \begin{table}
\begin{tabular}{c c c c} \hline
**Uncertainty Source** & \multicolumn{2}{c}{**Type of**} & \\
**Distribution** & **Divider** & \(\mathbf{u(x_{i})}\) \\ \hline Type A uncertainty & t-Student & \(\sqrt{10}\) & 0.2189 N \\ Output voltage by the NI 9237 & Rectangular & \(\sqrt{3}\) & 0.01173 mV \\ The conversion factor & Rectangular & \(\sqrt{3}\) & \(1.1547\times 10^{-6}\) mV/N \\ Offset voltage of the test load cell & Rectangular & \(\sqrt{3}\) & \(1.5083\times 10^{-4}\) mV \\ \hline \end{tabular}
\end{table} 
Table 3: Sources of uncertainty. Thus, the standard combined uncertainty for the thrust measurement is as shown below:  
\[u_{c\ thrust}=\sqrt{u(x_{1})^{2}+F^{2}\frac{u(x_{2})^{2}}{\left(V_{out\ eff}-V_{0} \right)^{2}}+F^{2}\frac{u(x_{3})^{2}}{C_{f}^{2}}+F^{2}\frac{u(x_{4})^{2}}{\left( V_{out\ eff}-V_{0}\right)^{2}}} \tag{13}\]  
where \(u(x_{1})\), \(u(x_{2})\), \(u(x_{3})\) and \(u(x_{4})\) are, respectively, the standard combined uncertainties due to Type A experimental data, the output voltage corrected by the NI 9237 module, the conversion factor, and the offset voltage of the test load cell. Based on the values presented in Table 4 and considering \(V_{out\ eff}=10.165\) mV, \(C_{f}=\) mV/N, and \(V_{0}=0.1306\) mV, we find \(u_{c\ thrust}=2.9466\) N, corresponding to 0.12 % of the mean of the test values. Equation 9 should be used to calculate the expanded uncertainty to determine the effective degrees of freedom, enabling the selection of an appropriate coverage factor. \[v_{eff\ thrust}=\frac{u_{c\ thrust}^{4}}{\frac{u(x_{1})^{4}}{9}+\frac{u(x_{2})^{4 }}{\infty}+\frac{u(x_{3})^{4}}{\infty}+\frac{u(x_{4})^{4}}{\infty}} \tag{14}\]  
For the calculated number of degrees of freedom, and according to the table in JCGM (2008) [2], it is possible to establish a coverage factor \(K=2\), for a confidence level of 95.45%. Therefore, the combined uncertainty is multiplied by the coverage factor (\(k\)) to obtain the estimated expanded uncertainty (\(U_{\exp thrust}\)) for a measurement chain with an amplified signal. \[U_{\exp thrust}=k\times u_{c\ thrust}=5.8932\ \mathrm{N}=0.24\ \% \tag{15}\]  
Table 4 presents the components of measurement uncertainty considered in this study, highlighting the main sources of variability that impact the precision of the results.