## 5 Numerical Examples
### Dimensional Synthesis of Watt II Six-Bar Slider-Crank Function Generator  
The prescribed positions that the Watt II six-bar slider-crank function generator should satisfy are given in Table 3, which shows the relative displacement of the output slider, \(\Delta p_{j}=p_{j}-p_{1}\), with respect to the relative input crank angle, \(\Delta\theta_{j}=\theta_{j}-\theta_{1}\), for \(i=2,\,\ldots,\,9\), each measured from the initial positions of the slider and input crank, respectively. The free choices were assumed as \(r_{1x}=0.12268\) and \(r_{1y}=0.87294\). For the given precision points, Equations (19) and (37) are solved by using the Bertini software package. The computation time was 3.42 h on a single node of Intel(r) Xeon(r) W-2245 [EMAIL_ADDRESS] GHz. Among 25,630 nonsingular solutions obtained, those that have less than 0.01% structural error are selected as feasible solutions. As a result, there were 37 mechanisms that have completely rotatable cranks and pass all the prescribed positions. Table 4 shows the number of the feasible mechanisms for each assembly configuration. The kinematic diagrams in their initial positions and slider displacements of two synthesized mechanisms are shown in Figure 3 and the corresponding solutions are listed in Table 5. \begin{table}
\begin{tabular}{c c} \hline \hline
**Configuration** & **Number of Feasible Mechanisms** \\ \hline
1 (Assembled by \(\alpha^{-},E_{y}^{+}\)) & 0 \\
2 (Assembled by \(\alpha^{-},E_{y}^{-}\)) & 1 \\
3 (Assembled by \(\alpha^{+},E_{y}^{+}\)) & 25 \\
4 (Assembled by \(\alpha^{+},E_{y}^{-}\)) & 11 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The number of feasible mechanisms for each configuration. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Precision Points (\(\boldsymbol{j}\))** & **1** & **2** & **3** & **4** & **5** & **6** & **7** & **8** & **9** \\ \hline \(\Delta\theta_{j}\) (deg) & 0 & 21 & 70 & 100 & 124 & 164 & 193 & 224 & 298 \\ \hline \(\Delta\eta_{j}\) & 0 & \(-0.49087\) & \(-1.45837\) & \(-1.69238\) & \(-1.77397\) & \(-1.77643\) & \(-1.67172\) & \(-1.42028\) & \(-0.13665\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Prescribed positions for the Watt II slider-crank function generator. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Configuration**} & \multicolumn{2}{c}{
\begin{tabular}{c} **Type of** \\ **Four-Bar** \\ **Mechanism** \\ \end{tabular} } & \multirow{2}{*}{**Solutions**} & \multirow{2}{*}{**Link Lengths**} & \multicolumn{2}{c}{**Orientations (Deg)**} \\ \cline{5-8}  & & & \(r_{2s}\) & \(1.83719848448098\) & \(r_{2}\) & \(2.66481499912287\) & \(\phi\) & \(46.4151406996215\) \\ \cline{3-8}  & & \(r_{2y}\) & \(1.93026959468645\) & & & & \\ \cline{3-8}  & & \(r_{3x}\) & \(2.29173698831327\) & & & & \\ \cline{3-8}  & & \(r_{3y}\) & \(-0.188601675140832\) & \(r_{3}\) & \(2.294948451051778\) & \(\alpha\) & \(-4.70463367830402\) \\ \cline{3-8}  & & \(r_{4x}\) & \(2.95628516701656\) & & & & \\ \cline{3-8}  & & \(r_{4y}\) & \(0.626085166776018\) & & & & \\ \cline{3-8}  & & \(r_{5x}\) & \(2.50589035665077\) & & & & \\ \cline{3-8}  & & \(r_{5y}\) & \(-2.02586988774379\) & & & & \\ \hline \multirow{8}{*}{3} & \multirow{8}{*}{Double-crank} & \(r_{2x}\) & \(-0.492252951916725\) & \(r_{2}\) & \(0.660020417350853\) & \(\phi\) & \(-138.229126001616\) \\ \cline{3-8}  & & \(r_{2y}\) & \(-0.439674860151526\) & & & & \\ \cline{3-8}  & & \(r_{3x}\) & \(-0.167431742713909\) & & & & \\ \cline{3-8}  & & \(r_{3y}\) & \(0.427810302149809\) & & & & \\ \cline{3-8}  & & \(r_{4x}\) & \(-0.66096098576403\) & & & & \\ \cline{3-8}  & & \(r_{4y}\) & \(0.712855804369779\) & & & & \\ \cline{3-8}  & & \(r_{5x}\) & \(-0.33301035320805\) & & & & \\ \cline{3-8}  & & \(r_{5y}\) & \(-1.366201942488444\) & & & & \\ \hline \hline \end{tabular}
\end{table}