# Accuracy Analysis of the Measurement of Centre of Gravity and Moment of Inertia with a Swing
## 3 Results
### Moment of Inertia (MI)  
The previous measurement of the centre of gravity based on different new stable equilibria caused by different added calibration weights. Instead of this stable conditions, the oscillating system is observed for the calculation of the moment of inertia (MI) based on Equation (4). The required input of the oscillation is the period \(T\), which is calculated based on the following approaches:  
* F1: simple cosine oscillation assuming a simple harmonic oscillator without the consideration of damping;
* F2: damped harmonic oscillator based on Equation (5);
* F3: similar to F2 but the length of the measurement is limited to a full period;
* F4: similar to F2 but a constant value is added similar to the approach shown in Equation (11). The empty swing is observed three times for each of the two pivot points P1 and P2 (Figure 2). Each capture time is at least 160 s. For the higher pivot point P2, the oscillation is too fast for the inertial inclinometer and the results from the Qualisys motion capturing system are used. As mentioned in Section 2.2, both systems capture the angle \(\alpha\) with a measurement frequency of 256 Hz. Table 5 results for the higher pivot point P2 and Table 6 for the lower one P1. For the fitting with the simple harmonic oscillation, the decay constant \(\lambda\) is not used and the initial amplitude \(A\) is smaller than for the other functions F2-F4. The latter is obvious because this fitting does not consider the reduction of the amplitude with the time and consequently delivers an average amplitude over the observation time. All other values are very stable and largely independent of the used fitting function. The period \(T\) is compared in detail in Table 7. This table also includes the difference with the averaged value for the period \(T\) for each P1 (7.2923 s) and P2 (2.8460 s). The maximum difference is 0.0159 s. Consequently, it can be summarised that the fitting results for the period of the oscillation \(T\) are largely independent of the chosen fitting function \(F1\) to \(F4\). Equation (4) introduces the connection between the constants and the values, which are depending on the specific pivot points. Those are the period \(T\) and the distance \(R\). The latter is equal to the distance \(d_{S}\) evaluated in Section 3.1. Table 8 summarises the input values and the calculation based on basic Equation (4) with a mass \(m\) of 42.280 kg and \(g\) equal 9.81 m s\({}^{-2}\). This value is called \(I_{Pcalc}\) and has as a reference point the individual pivot point P1 or P2. A comparison is possible by calculating the moment of inertia \(I\) around the CG based on Equation (6). The similar approach is used to calculate the MI for both pivot axis based on the individual measurement (one is measured and the other is calculated), which can be compared directly to the values calculated based on the theoretical approach (Table 1). The difference between the theoretical value and the measured are small and within an acceptable range. [PERSON] et al. [6] report an error of less than 3% and specify this as a very good result. \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline  & \multicolumn{3}{c}{**Run 1**} & \multicolumn{3}{c}{**Run 2**} & \multicolumn{3}{c}{**Run 3**} \\  & \(T_{\text{RER}}\) & \(\text{abs}(A)\) & \(\boldsymbol{\phi}\) & \(\Lambda\) & \(T_{\text{2R2}}\) & \(\text{abs}(A)\) & \(\boldsymbol{\phi}\) & \(\Lambda\) & \(T_{\text{2R2}}\) & \(\text{abs}(A)\) & \(\boldsymbol{\phi}\) & \(\Lambda\) \\ \hline F1 & 2.8455 & 2.816 & 1.111 & _not used_ & 2.8461 & 3.418 & \(-\)1.398 & _not used_ & 2.8462 & 3.693 & \(-\)0.086 & _not used_ \\ F2 & 2.8455 & 3.495 & 1.110 & 0.00299 & 2.8460 & 4.306 & \(-\)1.399 & 0.00321 & 2.8463 & 4.710 & \(-\)0.083 & 0.00339 \\ F3 & 2.8455 & 3.494 & 1.110 & 0.00299 & 2.8460 & 4.303 & \(-\)1.400 & 0.00320 & 2.8463 & 4.707 & \(-\)0.084 & 0.00338 \\ F4 & 2.8455 & 3.495 & 1.110 & 0.00299 & 2.8460 & 4.307 & \(-\)1.399 & 0.00321 & 2.8463 & 4.710 & \(-\)0.083 & 0.00338 \\ \hline \hline \end{tabular}
\end{table}