## 4 Kinematic Conceptual Design of a Six-Link VCR Engine Mechanism
### Minimization of Vertical Second Harmonic Acceleration
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & \multirow{2}{*}{**Case**} & \multirow{2}{*}{**Variation Number**} & \multicolumn{6}{c}{**Link Length**} \\ \cline{3-10}  & & & \(\mathbf{r_{2}}\) & \(\mathbf{r_{3}}\) & \(\mathbf{r_{4}}\) & \(\mathbf{r_{5}}\) & \(\mathbf{r_{6}}\) & \(\mathbf{r_{0y}}\) \\ \hline \multirow{3}{*}{1} & Decrease \(r_{3}\) with min. & 1 & 56 & 73.5 & 42 & 95.5 & 140 & \(-\)53 \\  & \(r_{2}\), \(r_{4}\), and max. \(r_{5}\) & : & 56 & : & 42 & 95.5 & 140 & \(-\)53 \\  & \(r_{2}\), \(r_{4}\), and max. \(r_{5}\) & 21 & 56 & 71.5 & 42 & 95.5 & 140 & \(-\)53 \\ \hline \multirow{3}{*}{2} & Decrease \(r_{6}\) with min. & 1 & 56 & 73.5 & 42 & 95.5 & 138 & \(-\)53 \\  & \(r_{2}\), \(r_{4}\), and max. \(r_{5}\) & : & 56 & 73.5 & 42 & 95.5 & : & \(-\)53 \\ \hline \multirow{3}{*}{3} & Decrease \(r_{0y}\) with min. & 1 & 56 & 73.5 & 42 & 95.5 & 140 & \(-\)53 \\  & \(r_{2}\), \(r_{4}\), and max. \(r_{5}\) & : & 56 & 73.5 & 42 & 95.5 & 140 & \(\vdots\) \\ \cline{1-1} \cline{3-10}  & \(r_{2}\), \(r_{4}\), and max. \(r_{5}\) & 21 & 56 & 73.5 & 42 & 95.5 & 140 & \(-\)55 \\ \hline \hline \end{tabular} The “-” in the table means the variation range as “-”. For example, the “1 : 21” and “73.5 : 71.5” for \(r3\) in Case 1 mean that the variation Number “1, 2, 3,..., 21” correspond “73.5, 73.4, 73.3,..., 71.5” of \(r3\).  
\end{table} 
Table 6: Link length variation cases for Step S3. Figure 8: With the minimum length of \(r_{4}\) and the maximum \(r_{5}\), influence on the maximum of vertical second harmonic acceleration sum of length change in: (**a**) \(r_{1}\) and \(r_{2}\) and (**b**) \(r_{3}\), \(r_{6}\), \(r_{0x}\), and \(r_{0y}\); influence on the stroke of length change in: (**c**) \(r_{1}\) and \(r_{2}\) and (**d**) \(r_{3}\), \(r_{6}\), \(r_{0x}\), and \(r_{0y}\) in the in-line four-cylinder engine. and fourth harmonic acceleration components in the in-line four-cylinder engine are shown and compared with those of the initial design. The maximum of the vertical second and fourth harmonic acceleration sums were decreased by 58.333% and 34.882%, respectively. The link geometry of the initially designed Stephenson III mechanism was compared with the final design with minimized vertical second harmonic acceleration in Figure 10.  
In Figure 11, the vertical harmonic acceleration components at the piston pin in the finally designed single-cylinder Stephenson III mechanism were compared with those of a conventional slider-crank mechanism whose crank length was 43 mm, offset was 0 mm, stroke was 86 mm, and piston pin position at TDC was 190 mm. The maximum vertical second harmonic acceleration at the piston pin in the final design of the Stephenson III mechanism was considerably lower than that of the slider-crank mechanism, each of which was calculated as 0.596 mm/rad\({}^{2}\) and 12.859 mm/rad\({}^{2}\). Even though the third and fourth acceleration components in the Stephenson III mechanism were calculated as 5.541 mm/rad\({}^{2}\) and 1.157 mm/rad\({}^{2}\), respectively, the vertical third harmonic acceleration would be cancelled out in the in-line four-cylinder engine and the vertical fourth harmonic acceleration was considerably lower than the second harmonic acceleration of the slider-crank mechanism. In Figure 12, the vertical second harmonic acceleration at each joint in the final design of the Stephenson III mechanism was compared with those of the slider-crank mechanism in the in-line four-cylinder engine configuration. In the case of the slider-crank mechanism, the peak value of the sum of the vertical second harmonic acceleration was 51.435 mm/rad\({}^{2}\), while the peak value of the overall or sum of the vertical second harmonic acceleration components of the Stephenson III mechanism was 14.963 mm/rad\({}^{2}\), which was about 29.091% of that for the slider-crank mechanism.