## 5 Analysis
### Expanding Optimization Limits for \(t_{3}\)  
As aforementioned, using the box-plots of the partial derivatives with respect to each variable, the volume values histogram of the counterweights, and the graphics of the relation between area and thickness, it is possible to obtain valuable information about the chosen limits of the optimization. The decision of modifying these limits depends on the mechanical characteristics of the whole mechanism.  
Taking the example when using only one counterweight (Appendix B) the recommendation after the analysis was to choose a higher limit of thickness \(t_{3}\), a new optimization process was executed with the limits:  
\[-0.40\ \mathrm{m}\leqslant x_{cn},\ y_{cn}\leqslant 0.40\ \mathrm{m} \tag{59}\]  
\[0.005\ \mathrm{m}\leqslant t_{cn}\leqslant 0.05\ \mathrm{m} \tag{60}\]  
The upper limit of \(t_{3}\) could be changed from 0.04 m to 0.05 m. Using this value and giving more importance to \(\beta_{ShF}\), the physical characteristics of Counterweight-3 should be:  
\[x_{c3}=-0.141021300369708\quad y_{c3}=-0.00665773774579448\quad t_{c3}=0.0207544 372440781\]  
Using this solution, allows the ShF and ShM to be reduced by 73.92% and 14.32%, respectively.  
Figure 8: Pareto front comparison.  
On the other hand, when the importance is given to \(\beta_{ShM}\), the physical characteristics of Counterweight-3 should be:  
\[x_{c3}=-0.1112378507983744\quad y_{c3}=0.091294702876245\quad t_{c3}=0.05\]  
Using this counterweight the ShM and ShF is reduced by 80.17% and 5.30%, respectively.  
Table 3 compares the optimization results obtained with the expanded limits and the original limits. It can be seen that expanding Counterweight-3 thickness upper limit, improves the ShM. Figure 9 shows the comparison of Pareto fronts before and after changing the optimization limits. It can be noticed that the thickness of Counterweight-3 has a greater influence on the optimization of the ShM of the mechanism. By increasing this limit, optimization results can be better.  
\begin{table}
\begin{tabular}{l l l l} \hline \hline  & & **ShF Optimization** & **ShM Optimization** \\ \hline \multirow{4}{*}{Using three counterweights} & Optimizing & \multirow{2}{*}{99.70\%} & \multirow{2}{*}{28.69\%} \\  & ShF & & \\ \cline{2-3}  & Optimizing & & \\  & ShM & & \\ \cline{2-3}  & Optimizing both & \multirow{2}{*}{54.82\%} & \multirow{2}{*}{57.03\%} \\  & ShF and ShM & & \\ \hline \multirow{4}{*}{Using counterweights} & Optimizing & \multirow{2}{*}{99.67\%} & \multirow{2}{*}{24.34\%} \\  & ShF & & \\ \cline{2-3}  & Optimizing & & \\ \cline{2-3}  & ShM & & \\ \cline{2-3}  & Optimizing & & \\ \cline{2-3}  & ShF and ShM & & \\ \hline \hline \end{tabular}
\end{table}
Table 2: Comparison between optimization results.  
\begin{table}
\begin{tabular}{l l l l} \hline \hline  & & **ShF Optimization** & **ShM Optimization** \\ \hline \multirow{2}{*}{Using 0.05m as \(t_{3}\) upper limit} & Optimizing & \multirow{2}{*}{73.92\%} & \multirow{2}{*}{14.32\%} \\  & ShM & & \\ \hline \multirow{2}{*}{Using 0.04m as \(t_{3}\) upper limit} & Optimizing & \multirow{2}{*}{5.30\%} & \multirow{2}{*}{80.17\%} \\  & ShM & & \\ \hline \multirow{2}{*}{Using 0.04m as \(t_{3}\) upper limit} & Optimizing & \multirow{2}{*}{78.74\%} & \multirow{2}{*}{0.42\%} \\  & ShM & & \\ \hline \hline \end{tabular}
\end{table}
Table 3: Comparison between optimization results when expanding the thickness upper limit of Counterweight-3.