# Design Procedure for Motion Profiles with Sinusoidal Jerk for Vibration Reduction
## 5 Experimental Results
The settling time of the proposed profile is identical to the moving time for the nominal case, as the vibration at the end of the motion falls within the specified allowable range;
* It can be seen from the dominant energy peaks in Figure 26a that the vibration mode at 8.81 Hz is mainly excited during the motion. The peak PSD magnitudes obtained using the trapezoidal, modified trapezoidal [16], seven-segment [23], and proposed optimized (\(r=1\)) profiles are 13.35 dB, \(-\)12.17 dB, \(-\)4.77 dB, and \(-\)21.80 dB, respectively. Notably, the energy peak is remarkably reduced with the optimized profile, supporting the finding of a more than 90% vibration reduction, as depicted in Figure 23. Moreover, the PSD of the excitation vibration with this profile is smaller than those with the other methods in the high-frequency domain, indicating the smoothness property of the profile;
* In the case of a modeling error, the performance of the proposed profile degrades but still remains superior to the other profiles, as depicted in Figure 24. Increasing the robustness degree (\(r\)) can enhance the insensitivity to parameter perturbation. The increment in the settling time due to the modeling error for the optimized profile with the \(r\) set to 1 is approximately one order of magnitude, while it remains unchanged with the \(r\) set to 3 (Figure 25b);
* While the developed profile demonstrates an outstanding capability of vibration reduction in lightly damped systems, a limitation is that this capability decreases when the damping ratio is significant. Therefore, for future work, it is crucial to improve the profile design by utilizing a complex domain transformation to address damped cases. On the whole, the obtained experimental results are in good agreement with the numerical results, clearly validating the superiority of the proposed profile optimization method in terms of robust vibration suppression for high-speed machines. This finding \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Motion Profile**} & \multicolumn{2}{c}{**Residual Vibration (m/s\({}^{2}\))**} & \multicolumn{2}{c}{**Settling Time (s)**} \\ \cline{2-6}  & **w/o Modeling Error** & **w/ Modeling Error** & **w/o Modeling Error** & **w/ Modeling Error** \\ \hline Trapezoidal velocity & 11.065 (ref) & 15.019 (ref) & 16.611 (ref) & 23.230 (ref) \\ [PERSON] et al. [16] & 0.720 (\(-\)93.5\%) & 10.386 (\(-\)30.9\%) & 3.653 (\(-\)78.0\%) & 20.733 (\(-\)10.8\%) \\ [PERSON] et al. [23] & 1.297 (\(-\)88.3\%) & 4.283 (\(-\)71.5\%) & 6.811 (\(-\)59.0\%) & 16.517 (\(-\)28.9\%) \\  & \(r=1\) & 0.293 (\(-\)97.4\%) & 1.643 (\(-\)89.1\%) & **1.121 (\(-\)93.3\%)** & 10.459 (\(-\)55.0\%) \\ Proposed & \(r=2\) & **0.150 (\(-\)98.6\%)** & 1.221 (\(-\)91.9\%) & 1.148 (\(-\)93.1\%) & 7.303 (\(-\)68.6\%) \\  & \(r=3\) & 0.292 (\(-\)97.4\%) & **0.207 (\(-\)98.6\%)** & 1.192 (\(-\)92.8\%) & **1.192 (\(-\)94.9\%)** \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Experimental comparison results of residual vibrations and settling times induced by different motion profiles. is beneficial for substantially enhancing the positioning accuracy and productivity in industrial applications.