## 3 Results and Discussion
### Calculation of Total Impulse Measurement Uncertainty  
As described in Equation (1), the total impulse is the integral of thrust over time (area under the thrust curve). This integration is performed using numerical methods, particularly within this acquisition system, using the trapezoidal rule. In the trapezoidal rule, the integration is given by  
\[l_{t}\approx\frac{h}{2}[T(t_{1})+2T(t_{2})+\ldots+2T(t_{n-1})+T(t_{n})] \tag{16}\]  
where \(h\) is the acquisition time interval between two consecutive samples, \(t_{1}\) is the first acquisition point, \(t_{n}\) is the nth acquisition point and \(n\) is the number of acquisition points. \begin{table}
\begin{tabular}{c c} \hline \hline
**Uncertainty Source** & **Contribution** \\ \hline Type A uncertainty & 0.55\% \\ The output voltage by NI 9237 & 93.44\% \\ The conversion factor & 5.99\% \\ The offset voltage of the test load cell & 0.02\% \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Contribution of each uncertainty source. \(T(t_{i})\) represents the thrust values yielded by the measurement system, each associated with an expanded uncertainty of 5.8932 N, as per Equation (15). Considering that the time measurement uncertainty is infinitesimal, \(h\) is a constant expressed by \(h=\frac{1}{f_{s}}\), where \(f_{s}\) is the sampling frequency. Applying the partial derivative of Equation (7) again to the numerical integration model expressed in Equation (16), the following sensitivity coefficients are obtained:  
\[\text{Coefficient }T(t_{1})\text{: }\frac{\partial I_{t}}{\partial T(t_{1})}= \frac{h}{2} \tag{17}\]  
\[\text{Coefficient T}(t_{i})\text{: }\text{for }i=2\text{ to }i=n-1\text{ }\frac{\partial I_{t}}{\partial T(t_{i})}=h \tag{18}\]  
\[\text{Coefficient T}(t_{n})\text{: }\frac{\partial I_{t}}{\partial T(t_{n})}= \frac{h}{2} \tag{19}\]  
Since the measurement uncertainty of \(T(t_{i})\) corresponds to that of the thrust, we have  
\[u_{c\text{ }impulse}=\sqrt{\left(\frac{h}{2}\right)^{2}\cdot u_{c\text{ } thrust}t^{2}+\frac{n-1}{2}\left(h\right)^{2}\cdot u_{c\text{ } thrust}t^{2}+\left(\frac{h}{2}\right)^{2}\cdot u_{c\text{ } thrust}t^{2}} \tag{20}\]  
\[u_{c\text{ }impulse}=\frac{u_{c\text{ } thrust}}{f_{s}}\times\sqrt{\frac{2n-3}{2}} \tag{21}\]  
Additionally, we can consider that the uncertainty due to the error in the numerical integration model can be estimated by the following formula [19]:  
\[u_{num.\text{ }integration}=\frac{\Delta t^{3}}{12n^{2}}\times f^{\prime\prime }\left(\xi\right) \tag{22}\]  
where \(f^{\prime\prime}\left(\xi\right)\) is the second derivative of function \(I_{t}\) evaluated at point \(\xi\) in interval \(\Delta t[t_{1},t_{n}]\). In the case of discrete data, directly estimating \(f^{\prime\prime}\left(\xi\right)\) can be challenging. A common approach is to use a conservative estimate for the error based on the smoothness of the curve. As the sampling frequency is high, meaning the number of sampled points is sufficiently large, resulting in a reduced spacing h, the numerical integration methods tend to produce very close results. When the number of subdivisions \(n\) reaches the order of thousands, the truncation error in the trapezoidal method decreases substantially due to the \(\frac{1}{n^{2}}\) term in its error formula, reducing the error to around \(10^{-5}\). Here, a sampling rate of 25,000 Hz was used, ensuring high precision in the results. Considering the lower computational cost and the simplicity of implementation, the trapezoidal method was chosen to integrate thrust curves in static tests of rocket motors. This choice proved to be appropriate as, at high frequencies, the difference between integration methods became negligible, while computational efficiency was preserved. As a result, the combined standard uncertainty is primarily dominated by the measurement uncertainty of Equation (21).