# Analysis of Underlapped Symmetrically Ported Valve-Controlled Asymmetric Cylinder Drive
## 3 Materials and Methods
### Root Form and Identification  
A general form of roots can be depicted as below in Equation (43):  
\[\left\{ \begin{array}{l}X_{1}=-\frac{a}{4}+\frac{(U+V)^{\frac{1}{2}}}{2}-\frac{(2 U-V-W)^{\frac{1}{2}}}{2}\\ X_{2}=-\frac{a}{4}-\frac{(U+V)^{\frac{1}{2}}}{2}-\frac{(2U-V+W)^{\frac{1}{2}}}{2} \\ X_{3}=-\frac{a}{4}+\frac{(U+V)^{\frac{1}{2}}}{2}+\frac{(2U-V-W)^{\frac{1}{2}}}{2} \\ X_{4}=-\frac{a}{4}-\frac{(U+V)^{\frac{1}{2}}}{2}+\frac{(2U-V+W)^{\frac{1}{2}}}{2} \end{array} \right. \tag{43}\]  
where  
\[ \begin{array}{l}U=\frac{a^{2}}{4}-\frac{2b}{5}\\ V=\frac{1}{5}+\frac{1}{3d}\\ W=\frac{\left(a^{2}-4 ab+8c\right)}{4(U+V)^{\frac{1}{2}}}\end{array}  \tag{44}\]  
where  
\[ \begin{array}{l}Z_{0}=b^{2}-3 ac+12d\\ Z_{1}=2b^{3}-9 abc+27a^{2}d+27c^{2}-72 bd\\ L=\left[\frac{Z_{1}+\left(Z_{1}{}^{2}-4Z_{0}\right)^{\frac{1}{2}}}{2}\right]^{ \frac{1}{3}}\end{array}  \tag{45}\]  
Actually, there is only one value of factor \(X\) with a given set of parameters. However, there are four roots from a quartic equation and the value of \(X\) is one of them. To identify the correct root, the method is depicted as the following steps:  
1. Numerically solving the \(X\) factor in the underlap region. 2. Identify one of the roots in a certain valve displacement in the underlap region corresponding to the same value from the numerical solution. 3. Repeat step 1 and 2 with different area ratio \(\gamma\)
4. Repeat step 1, 2, and 3 with different underlap regions. 5. Estimate the final analytical solution of the \(X\) factor. Starting with the parameter from [PERSON]'s valve-controlled asymmetric cylinder drive system, for the area ratio \(\gamma=11.43/5.8=1.9707\) and underlap region from 0.0045 mm to \(-0.0045\) mm, root identification is listed in Table 1. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Valve Disy \(X_{s}\) (mm)** & **X Factor** & **Root** \\ \hline \(-0.0045\) & 3.883617272 & \(X_{4}\) \\ \(-0.004404\) & 3.867534491 & \(X_{4}\) \\ \(-0.004296\) & 3.848941124 & \(X_{4}\) \\ \(-0.004101\) & 3.814000308 & \(X_{4}\) \\  & \(\vdots\) & \\  & \(\vdots\) & \\ \(-0.0009\) & 2.963117621 & \(X_{4}\) \\ \(-0.000753\) & 2.911892307 & \(X_{4}\) \\ \(-0.000735\) & 2.905859937 & \(X_{1}\) \\ \(0\) & 2.637188916 & \(X_{1}\) \\ \(0.0012\) & 2.172272301 & \(X_{1}\) \\ \(0.00255\) & 1.65272018 & \(X_{1}\) \\  & \(\vdots\) & \\  & \(\vdots\) & \\
0.004101 & 1.119970977 & \(X_{1}\) \\
0.0042 & 1.0894822 & \(X_{1}\) \\
0.004302 & 1.055863937 & \(X_{1}\) \\
0.00441 & 1.026380398 & \(X_{1}\) \\
0.0045 & 1 & \(X_{1}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Identifying the root from numerically solved results when the area ratio is 1.9707. A new phenomenon can be noticed: the \(X\) factor values in the underlap region contain two roots of Equation (25), \(X_{1}\) and \(X_{4}\). When valve displacement is from \(-0.0045\) mm to \(-0.00075\) mm, the value of the \(X\) factor is the root \(X_{4}\), and when valve displacement is from \(-0.00074\) mm to \(0.0045\) mm, the value of the \(X\) factor is the root \(X_{1}\). All four root values in the underlap region are described as in Figure 10a, different line styles corresponding to different roots. The values of the roots in the underlap are compared with the numerically solved \(X\) values, as in Figure 10b. It can be noticed that a part of \(X_{4}\) and a part of \(X_{1}\) are merged into the numerically solved curve, and they meet at some specific point. Another interesting phenomenon can be observed: that some parts of the root curve can be combined with parts of other root curves to generate a smoothed curve. The point where \(X_{4}\) and \(X_{1}\) curves meet may change with some parameters. We can change the form of valve opening area \(A_{v1}\) and \(A_{v2}\) in Equation (25) into the equation below. \[