## 5 Discussion  
The conceptual design of a new VCR engine may depend on trial and error to find the feasible kinematic structure. The purpose of this research was to propose a simple method to determine a feasible mechanism and its dimensions that would yield a nearly minimized sum of the vertical second harmonic acceleration at joints without going through an optimization procedure. For the validation of the proposed method, a constrained nonlinear optimization was carried out using 'fmincon' solver of MATLAB optimization tool box to minimize the maximum sum of the second harmonic acceleration at each joint with the design specification given in Table 2 and the Grashof criteria for the crank-rocker four bar as the constraints. The computational time of the optimization was 9.6 s using a computer (Intel@i7 Quad CPU 1.8 GHz, 8 GB RAM) with Windows 10 (64 bit) operating system. The results of the optimization were compared with that of the proposed method in Table 8 and Figure 13. As shown in Table 8, the dimensions determined by the proposed method and the optimization were slightly different, because the selected variables in the proposed procedure were the link lengths except \(r_{1}\) and \(r_{0x}\), while the optimization considered all the link lengths as variables. Even though the optimization method yielded optimum dimensions when a mechanism to be optimized was selected, it did not provide information that could be used to select a feasible mechanism among many candidates. The proposed method, on the other hand, can be applied at the kinematic conceptual design stage to determine a feasible mechanism and its dimensions simply by plotting the vertical second harmonic acceleration component at each joint and the influence of link length changes. The influence of link length changes on the second harmonic acceleration can be utilized at the final layout design stage for a minute adjustment of link lengths and optimization as well. Figure 13: Vertical second harmonic acceleration components at joints of the Stephenson III mechanism in the four-cylinder engine: (**a**) proposed method and (**b**) optimization using commercial software. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & & **Unit** & \begin{tabular}{c} **Initial** \\ **Design** \\ \end{tabular} & \
\begin{tabular}{c} **Result of** \\ **Proposed Method** \\ \end{tabular} &
\begin{tabular}{c} **Result of** \\ **Optimization** \\ \end{tabular} & **Effect (\%)** \\ \hline \multirow{8}{*}{Link} & \(r_{1}\) & mm & 25.000 & 25.000 & 24.95798 & - \\  & \(r_{2}\) & mm & 57.000 & 56.000 & 56.0000 & - \\  & \(r_{3}\) & mm & 73.500 & 73.500 & 73.4895 & - \\  & \(r_{4}\) & mm & 44.000 & 42.000 & 42.0000 & - \\  & \(r_{5}\) & mm & 93.500 & 95.500 & 95.49999 & - \\  & \(r_{6}\) & mm & 140.000 & 140.000 & 138.000 & - \\  & \(r_{0x}\) & mm & 78.000 & 78.000 & 80.0000 & - \\  & \(r_{0y}\) & mm & \(-\)53.000 & \(-\)53.900 & \(-\)55.00000 & - \\ \hline \multirow{4}{*}{Result} & Stroke & mm & 85.994 & 86.018 & 85.975 & - \\  & Piston pin position & mm & 197.391 & 185.124 & 184.424 & - \\ \cline{1-1}  & Max. 2 nd harmonic acceleration sum & mm/rad\({}^{2}\) & 35.911 & 14.963 & 13.773 & \(-\)7.940 \\ \cline{1-1}  & Max. 4 th harmonic acceleration sum & mm/rad\({}^{2}\) & 10.714 & 6.977 & 7.114 & +1.997 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Comparison of the results of the proposed method and the constrained optimization. For the completion of the kinematic design of a VCR engine, the control function needs to be considered. The proposed procedure can be initially applied to determine the dimensions of the mechanism for a specific compression ratio at which the vertical second harmonic acceleration needs to be taken into account more seriously. In order to vary the compression ratio, the ground pivot, \(P_{3}\), in Figure 10b needs to be moved to a new position.