## 2 Materials and Methods: Architectural Configurations for the Case Study
### The Hydraulic System
#### 2.1.3 Theoretical Analysis of a Valve-Controlled Circuit  
System sizing requires actual forces and velocities and consequently pressures and flow rates. This section contains the theoretical analysis required to calculate these quantities. The simplest control circuit that can be used for a linear actuator combines the differential cylinder with an area ratio of 2:1 with a standard 1:1:1:1 directional control valve. Let us adopt the nomenclature from Figure 4: the flow rate through the rear chamber and flow rate through the front chamber of the actuator are defined as \(Q_{A}\) and \(Q_{B}\), respectively. \(p_{A}\) and \(p_{B}\) denote the relative pressures occurring in the rear and front chambers of the cylinder, respectively. \(p_{P}\) is the pressure occurring at the outlet of the pumping unit. The fluid pressure in the reservoir and pressure tank is denoted by \(p_{T}\). Since the oil pressure in the tank is generally equal to atmospheric pressure, the pressure \(p_{T}\) is assumed to be zero as a value relative to atmospheric pressure. For a differential cylinder, the area ratio \(\alpha\)  
Figure 3: Flowchart of cylinder sizing. \begin{table}
\begin{tabular}{c l l} \hline
**Circuit Element** & **Parameter** & **Value** \\ \hline \multirow{3}{*}{Fluid} & Code & Castrol serie HYSPIN ZZ 46 \\  & Fluid type & Mineral oil \\  & Viscosity & ISO VG 46 \\ \hline \multirow{6}{*}{Cylinder} & Code & Parker 140 MF3 MMAXRN23M300M1133 AOAO \\  & Type & Double acting \\  & Bore & 140 (mm) \\ \cline{1-1}  & Rod diameter & 100 (mm) \\ \cline{1-1}  & Stroke & 300 (mm) \\ \cline{1-1}  & Fixing & Front flange with feet and fixed load with rigid guide \\ \cline{1-1}  & Area Ratio & 1.96 \\ \hline \end{tabular}
\end{table} 
Table 5: Fluid and cylinder: designations and technical data. is defined as \(\alpha=\frac{A_{P}}{A_{T}}\) and the flow rates for the two chambers in the exit stroke have the values given in Equation (1). For the 4-way directional valve used (P = pressure, T = discharge, A = first port; B = second port), 1:1:1:1 means that for the paths AT:PA:PB:BT the pressure drop is equal to \(\Delta p_{n}\) when 100% of the nominal flow \(\Delta Q_{n}\) passes through it. The value 1 stands for 100%, another value means another percentage, e.g., 0.5 for 50%. In the following, the pressure drops at the DCV, the pressures generated in the rear and front chambers of the cylinder, the thrust and traction forces that can be exerted by the rod, and the maximum speed it can reach during the outward and return phases are determined at the theoretical level. In the analysis, both the pressure and the translational velocities of the rod are assumed to be constants with respect to the ejection and the return. To distinguish the parameters related to the outward stroke from those related to the inward stroke, the superscripts \"\(\alpha\)\" (outward) and \"\(\dot{1}\)\" (inward) are used. \[Q_{A}^{o}=vA_{P}\hskip 28.452756 ptQ_{B}^{o}=\frac{Q_{A}^{o}}{\alpha} \tag{1}\]  
The relationship between the flow rate traversing a DCV path and the pressure drop occurring on the path itself is expressed by Equation (2). \[Q_{x}=Q_{n}\sqrt{\frac{\Delta p_{x}}{\Delta p_{n}}} \tag{2}\]  
Consequently, the pressure drops on the PA and BT paths can be expressed using Equations (3) and (4). \[\Delta p_{PA}=p_{P}-p_{A}^{o}=\Delta p_{n}\bigg{(}\frac{Q_{A}^{o}}{Q_{n}} \bigg{)}^{2}=\Delta p_{n}\bigg{(}\frac{vA_{P}}{Q_{n}}\bigg{)}^{2} \tag{3}\]  
\[\Delta p_{BT}=p_{B}^{o}-p_{T}=p_{B}^{o}=\Delta p_{n}\bigg{(}\frac{Q_{B}^{o}}{Q _{n}}\bigg{)}^{2}=\Delta p_{n}\bigg{(}\frac{vA_{P}}{\Delta Q_{n}}\bigg{)}^{2} \tag{4}\]  
Figure 4: Symbols for theoretical analysis of the system. For a cylinder with area ratio 2, the ratio between the pressure drops between ports P and A and B and T is \(\frac{\Delta p_{PA}}{\Delta p_{BT}}=\alpha^{2}=4\). From Equations (3) and (4), the pressures in the rear and front chambers of the actuator can be derived (Equations (5) and (6)).