# Design and Calibration of a Tri-Directional Contact Force Measurement System
## 5 Dynamic Loading Experiment
### Results of the Dynamic Loading Experiment  
To identify the threshold frequency, a preliminary experimental modal testing was performed. Excitation was provided by the shaker, while response of the system was measured by three accelerometers placed on the chamfered block oriented along X, Y, and Z directions. Frequency Response Function (FRF) curves (Figure 10) were obtained to determine the natural frequencies of the measurement system and to determine the maximum allowable excitation frequency.  
Based on the results in Figure 10, the first natural frequency of the system is around 130 Hz; as a consequence excitation frequencies of 100 Hz and 200 Hz were defined. At 100 Hz the system dynamics is not expected to affect the measurement accuracy, while at 200 Hz, significant periodic inertia forces of the system are expected.  
After determining the natural frequencies of the setup, dynamic loading experiment was performed by simultaneous application of a harmonic force in the (_x,y_), while a static force was applied in the \(z\) direction. The mean absolute error of both measured and decoupled forces was computed over 100 cycles and results are shown in Table 3, at the two selected excitation frequencies and two different load cases.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multicolumn{2}{c}{**Applied Forces (N)**} & \multicolumn{6}{c}{**Mean Absolute Error (N)**} \\ \hline
**Freq** & **Amp** & **Static** & \multicolumn{2}{c}{**Measured Force Outputs**} & \multicolumn{2}{c}{**Decoupled Applied Forces**} \\ \hline
**Hz** & \(F_{x}\)**\(F_{y}\)** & \(F_{z}\) & \(F_{1}\) & \(F_{2}\) & \(F_{3}\) & \(\widetilde{F}_{x}\) & \(\widetilde{F}_{y}\) & \(\widetilde{F}_{z}\) \\ \hline
100 & 6.5 & 7.8 & 0.2218 & 0.4966 & 0.3642 & 0.1630 & 0.2269 & 0.1403 \\
200 & 6.5 & 7.8 & 0.3008 & 1.0378 & 0.7806 & 0.3377 & 0.7563 & 0.4419 \\ \hline
100 & 12 & 17.8 & 0.5290 & 1.1998 & 0.7268 & 0.3244 & 0.1880 & 0.1732 \\
200 & 12 & 17.8 & 0.6468 & 1.6479 & 1.1920 & 0.5984 & 0.8607 & 0.6917 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Dynamic loading test result comparison and effect due to variation in excitation frequency.  
Figure 10: Frequency Response Function (FRF) plot of the tri-directional force measurement calibration setup.  
Furthermore, in this case, the application of the decoupling method to the raw measurements generally improves the accuracy of the measurement. In detail, in the first load case, at both the excitation frequencies, a relative error reduction of 61% and 43% is observed, respectively, except for Sensor 1 at 200 Hz where a marginal increase (12%) is observed. A similar behavior is observed for the second load case, with a relative error reduction up to 84% and 47%, respectively. As expected, at 200 Hz the mean absolute error is larger than at 100 Hz for both the load cases.  
A more detailed investigation about the effect of the excitation frequency on the measurement accuracy is shown in Figure 11, where the periodic measured and decoupled forces are plotted vs. the applied loads. The plots confirm the higher accuracy of the measurements at 100 Hz. It is worth noting that at 200 Hz, the decoupling method is not able to remove the artificial periodic component measured by Sensor 3.  
Figure 11: Decoupling method applied in dynamic loading experiment at **(a)** 100 Hz and **(b)** 200 Hz.  
Having ruled out the 200 Hz frequency, due to the inaccuracy of the resulting measurements, the effect of the force amplitude on the accuracy of the measurement was performed at 100 Hz only. Results in Table 4 show that the initial measurement error increases as the amplitude of the excitation forces increases. Nevertheless, the decoupling method stays effective for all the combination of external inputs tested in the experimental campaign.