# Analysis of Underlapped Symmetrically Ported Valve-Controlled Asymmetric Cylinder Drive
## 2 Theory
### The Actuator Model  
The actuator model connects the valve and load model, which accepts the information of flow and pressure as input, then outputs force to the load model. The types of hydraulic actuators are mainly symmetric cylinder, asymmetric cylinder, and hydraulic motor. Due to the focus of this research, the asymmetric cylinder is the only one under analysis. Whether the asymmetric cylinder is in the extending or retracting state, the flow in its chamber consists of input flow, displacement flow, and compressibility flow. However, under different operational states, the flow balance equations in the different chambers are not the same. For the cylinder extending state:  
\[Q_{1}=Q_{D1}+Q_{C1} \tag{10}\]  
\[Q_{2}=Q_{D2}-Q_{C2} \tag{11}\]  
For the cylinder retracting state:  
\[Q_{2}=Q_{D2}+Q_{C2} \tag{12}\]  
\[Q_{1}=Q_{D1}-Q_{C1} \tag{13}\]  
Figure 5: Theoretical and actual situation of a zero-lapped valve [6]. Figure 6: Flow gain model block diagram. where  
\[Q_{\text{D1}}=v\cdot A_{1} Q_{D2}=v\cdot A_{2} \tag{14}\] \[Q_{\text{C1}}=\frac{sP_{1}\cdot V_{1}}{B} Q_{\text{C2}}=\frac{sP_{2}\cdot V_{2}}{B} \tag{15}\]  
A is the area of the cylinder piston side area or rod side area [m\({}^{2}\)], \(v\) is the cylinder velocity [\(\frac{m}{s}\)], \(V\) is the cylinder chamber volume [m\({}^{3}\)], \(P\) is the cylinder chamber pressure [\(\frac{N}{m^{2}}\)] and \(B\) is the fluid Bulking Modulus [\(\frac{N}{m^{2}}\)]. The subscript 1 and 2 indicate the piston side chamber and rod side chamber, respectively. A generalised definition of these equations is proposed by [PERSON] [8], which uses effective parameters to describe the system behaviours rather than analyse them separately in different cylinder chambers. The generalised equations are depicted as follows:  
\[ \begin{array}{l}P_{LE}=\frac{P_{1}A_{1}-P_{2}A_{2}}{A_{2}} Q_{LE}=\frac{Q_{1}+Q_{2}}{2}\\ A_{E}=\frac{A_{1}+A_{2}}{2}Q_{DE}=\frac{Q_{D1}+Q_{D2}}{2}\\ Q_{CE}=\frac{Q_{\text{C1}}+Q_{CE}}{2}=Q_{LE}-Q_{DE}\end{array}  \tag{16}\]  
Substituting these effective parameters into the flow and pressure equations reveals the following:  
For cylinder extending (\(A_{V2}\) = 0 and \(A_{V1}\) > 0), the service line pressures can be rewritten as follows [8]:  
\[\left(P_{1}\right)_{ext}=\frac{P_{s}+\gamma^{2}\frac{1}{2}(1+\gamma)P_{LE}}{1+ \gamma^{3}} \tag{17}\]  
\[\left(P_{2}\right)_{ext}=\frac{\gamma P_{s}-\frac{1}{2}(1+\gamma)P_{LE}}{1+ \gamma^{3}} \tag{18}\]  
For cylinder retracting (\(A_{V2}\) > 0 and \(A_{V1}\) = 0), the service line pressures can be rewritten as follows [8]:  
\[\left(P_{1}\right)_{ret}=\frac{\gamma^{2}P_{s}+\gamma^{2}\frac{1}{2}(1+\gamma )P_{LE}}{1+\gamma^{3}} \tag{19}\]  
\[\left(P_{2}\right)_{ret}=\frac{\gamma^{3}P_{s}-\frac{1}{2}(1+\gamma)P_{LE}}{1 +\gamma^{3}} \tag{20}\]  
where \(\gamma\) is the area ratio of the cylinder. [PERSON] [8] introduced the \(X\) factor to unify the expression for \(P_{1}\), \(P_{2}\) and utilised generalised concepts in the component linking method. The \(X\) factor value depends on the region of the spool operation: \(X\) = 1 for cylinder extending state and \(X\) = \(\gamma^{2}\) for cylinder retracting state. The service line pressure \(P_{1}\) and \(P_{2}\) can be written as follows:  
\[P_{1}=\frac{XP_{s}+\gamma^{2}\frac{1}{2}(1+\gamma)P_{LE}}{1+\gamma^{3}} \tag{21}\]  
\[P_{2}=\frac{X\gamma P_{s}-\frac{1}{2}(1+\gamma)P_{LE}}{1+\gamma^{3}} \tag{22}\]  
\[P_{LE}=\bigg{(}\frac{4B}{sV_{TE}}\bigg{)}Q_{CE} \tag{23}\]  
where \(\gamma\) is the area ratio, \(V_{TE}\) is effective total volume [m\({}^{3}\)], \(s\) is the Laplace operator, and \(P_{LE}\) is average pressure. When \(X\) = 1 for cylinder extending state, \(X\) = \(\gamma^{2}\) for cylinder retracting state. \[V_{TE}=\frac{2A_{E}}{A_{1}}\Bigg{[}\frac{V_{1}+\left(\frac{A_{2}}{A_{1}} \right)^{2}V_{2}}{1+\left(\frac{A_{2}}{A_{1}}\right)^{3}}\Bigg{]} \tag{24}\]The \(X\) factor value is known and clear when the valve is outside the underlap region.