# A Critical Analysis of Flow-Compensated Hydrostatic Single Rod Actuators: Simulation Study

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## Abstract

**:**

## 1. Introduction

_{1}and C

_{2}, connect the cap and rod sides of the circuit—indicated by the subscripts $p$ (piston) and $a$ (annulus), respectively—to the charge pressures, ${p}_{\mathrm{c}1}$ and ${p}_{\mathrm{c}2}$. Valves C

_{1}and C

_{2}may be combined into a single valve, C, and/or connected to a single charge pressure, ${p}_{\mathrm{c}}$. The compensation valves, C

_{1}and C

_{2}, are activated by external signals, ${s}_{\mathrm{c}1}$ and ${s}_{\mathrm{c}2}$, the nature of which depends on the technology being employed (hydraulic or electric). Likewise, signals ${s}_{\mathrm{p}}$ and ${s}_{\mathrm{a}}$ activate the inline valves, V

_{p}and V

_{a}, which may be present at some configurations. Note that the presence of valves V

_{p}and V

_{a}creates a pressure differential between the pump and the cylinder ports, i.e., ${p}_{\mathrm{p}0}\ne {p}_{\mathrm{p}}$ and ${p}_{\mathrm{a}0}\ne {p}_{\mathrm{a}}$. Conduit pressure losses are disregarded in our analysis, so that the pressure differentials between the pump and the cylinder cease to exist in the absence of valves V

_{p}and V

_{a}, as is the case in a number of circuit designs. The piston and annulus areas of the differential cylinder are ${A}_{\mathrm{p}}$ and ${A}_{\mathrm{a}}$, respectively. The piston-to-annulus area ratio is defined as $\alpha ={A}_{\mathrm{p}}/{A}_{\mathrm{a}}$. Finally, the external force is represented as $F$, and the friction force, ${F}_{\mathrm{f}}$, is given by [3].

_{p}and V

_{a}are not present, as in Figure 1a,c,d, we have that ${p}_{\mathrm{p}}={p}_{\mathrm{p}0}$ and ${p}_{\mathrm{a}}={p}_{\mathrm{a}0}$. Equation (3) then reduces to

## 2. Circuit with Pilot-Operated Check Valves

_{1}and C

_{2}depend on the operational quadrant and is given in Table 2.

_{1}and C

_{2}, in the circuit shown in Figure 5a are simultaneously open or closed [7,8,9]. In such case, it is clear that the circuit will not operate properly. Here, we extend the definition of “critical region” to encompass the area on the velocity versus external force diagram in Figure 3a, where valves C

_{1}and C

_{2}are incorrectly open or closed. By studying the circuit in Figure 5a, we identify that the correct behaviour of valves C

_{1}and C

_{2}should be as indicated on the second column of Table 2. However, this is not what has been observed and reported in practice, as will be fully described by performing a numerical simulation on the circuit in Figure 5a. Our starting point is to consider Equations (2) and (4). We rewrite these equations for the particular case where ${V}_{14}={V}_{58}=V$

_{1}and $p={p}_{\mathrm{a}}$ for valve C

_{2});$C$ is the valve coefficient, taken as $C=3.0\mathrm{L}/\left(\mathrm{min}.{\mathrm{bar}}^{1/2}\right)$, so that a $3\mathrm{L}/\mathrm{min}$ flow through the valve creates a pressure drop of $1.0\mathrm{bar}$. This is an average value but otherwise, quite reasonable and stands between the very low pressure drop of typical check valves (less than $1.0\mathrm{bar}$, for this flow magnitude) and the relatively higher pressure drops of directional valves (around $4.0\mathrm{bar})$, found in industrial manufacturer catalogues (see, for example, reference [17]). Under the assumption that valve channels can be treated as orifices, we might as well understand that the flow regime is turbulent within the valves.

_{1}and C

_{2}, we follow reference [13] where the opening conditions for valve C

_{1}are defined below:

_{2}

_{1}being opened and valve C

_{2}closed, as seen in Figure 6 (the status of valves C

_{1}and C

_{2}can be inferred by the presence of the flows ${Q}_{\mathrm{c}1}$ and ${Q}_{\mathrm{c}2}$). Note that, shortly after the beginning of the second quadrant, both flows, ${Q}_{\mathrm{c}1}$ and ${Q}_{\mathrm{c}2}$, are non-zeros, indicating that valves, C

_{1}and C

_{2}, remain simultaneously open, during the whole simulation. The practical implication is that the charge pump is driving the cylinder alongside with the main pump, meaning that the cylinder is not exclusively controlled by the pump as it extends.

_{1}and C

_{2}, are independently activated by either the cap- or rod-side pressures or by the charge pressure. These pressure signals can be combined in such a way that both valves may open or close at the same time or remain incorrectly open. Secondly, and most importantly, cap- and rod-side pressures do not seem to precisely indicate the correct moment when the compensation flow should be diverted to either sides of the circuit. Instead, it is clear that a minimum pressure level must be reached before the valves operate as expected. Based on these observations, some modifications have been proposed to improve the above circuit performance. We will analyse these modifications next.

## 3. Modified Versions of the Circuit with Pilot-Operated Check Valves

_{1}and C

_{2}are simultaneously open. We have already discussed this point in the previous section, where we extended this definition to include the incorrect opening or closing of the valves (see Table 2). To keep with the original definition given in [10], we focus on the interval where valves C

_{1}and C

_{2}are simultaneously open. Assuming that both valves open in piloted mode, we obtain the cap and rod-side pressures from Equation (6), as follows:

_{1}open in piloted mode, is

_{2}can be obtained by substituting ${p}_{\mathrm{a}}$, given by the second equation in (14) into the first inequality in (13):

_{1}and C

_{2}are open, we have that ${p}_{\mathrm{p}}={p}_{\mathrm{a}}={p}_{\mathrm{c}}$;

_{2}. This becomes evident when we note that the term multiplying ${p}_{\mathrm{c}}$ in Equation (16), $\left({K}_{\mathrm{p}}-1\right)$, is greater than zero. Likewise, since $\left(1-{K}_{\mathrm{p}}\right)<0$ in Equation (15), the upper limit, ${F}_{\mathrm{C}1}$, can also be reduced by increasing the charge pressure. We conclude, therefore, that the “critical region” would become narrower for higher values of ${p}_{\mathrm{c}}$. Alternatively, we could change the charge pressures for valves C

_{1}and C

_{2}separately.

_{1}remains open from point

**a**, which is the point where quadrants shift, according to the division in Figure 3a, and remains open past point

**b**, where valve C

_{2}is also activated. As a result, at least we conclude that C

_{1}and C

_{2}are not following the expected behaviour after point

**b**is reached. In addition, if the diagram in Figure 3a represents the correct quadrant division, the compensation valves are incorrectly open after point

**a**is reached (see Table 2). This has cast some doubts about the correct quadrant representation of Figure 3a and attempts have been made to draw diagrams on the force-velocity plane that are able to catch the quadrant shifting points (see, for example, references [7,10]).

#### 3.1. Circuit Using Pilot-Operated Check Valves with Different Charge Pressures

_{1}and C

_{2}independently. Figure 10 shows one way of creating a circuit with two different charge pressures, ${p}_{\mathrm{c}1}$ and ${p}_{\mathrm{c}2}$, with ${p}_{\mathrm{c}2}>{p}_{\mathrm{c}1}$ [10]. The idea is to reduce the critical region in Figure 8, by increasing the lower limit, ${F}_{\mathrm{C}2}$ (shifting it to the right). Practically, a pressure reducing valve, R, has been added between the charge circuit and valve C

_{1}.

_{1}and C

_{2}, are simultaneously open throughout the cylinder motion. Similar to what we saw in Figure 6, the cylinder extends uncontrolled in both quadrants.

#### 3.2. Circuit Using Inline Flow Throttling

_{p}is retrofitted into a proportional valve where the flow resistance changes with the pressure differential, $\left({p}_{\mathrm{p}0}-{p}_{\mathrm{a}}\right)$. When $\left({p}_{\mathrm{p}0}-{p}_{\mathrm{a}}\right)\ne 0$, valve V

_{p}gradually opens the passage between the pump and the cylinder cap-side. When $\left({p}_{\mathrm{p}0}-{p}_{\mathrm{a}}\right)=0$, valve V

_{p}centres, cutting the communication between pump and cylinder. Note that, due to throttling, ${p}_{\mathrm{p}0}>{p}_{\mathrm{p}}$ during the entire cylinder extension. Likewise, ${p}_{\mathrm{p}0}<{p}_{\mathrm{p}}$ during cylinder retraction. As a result, throttling losses are introduced throughout the cylinder motion, just like in a conventional valve-controlled actuator.

_{p}and ${R}_{\mathrm{Q}}$ is the flow resistance.

_{1}closed and valve C

_{2}open in piloted mode. The in-line resistance created by valve V

_{p}assures the correct behaviour of check valves during the first-quadrant of operation. Second quadrant operation, however, is poor, as will be shown next.

_{a}is not present (see Figure 2), the equation for $d{p}_{\mathrm{a}0}/dt$ was disregarded in Equation (3). With these considerations, the modelling equations become

_{p}(Figure 14b). Note that we are simulating the best-case scenario, where the resistance at valve V

_{p}remains constant and at a low value.

_{p}needs to be sized accordingly to the flow, and may become bulky and heavy. In such a case, the dynamics of the valve (not simulated in this paper) plays a role on the valve performance. The quadrant division in Figure 14 has been made by observing the sign of the external force (not shown in the figure), according to Figure 3a.

_{2}is closed by the rod-side pressure, ${p}_{\mathrm{a}}$ (Figure 15b). However, shortly after the cylinder starts moving, pressure ${p}_{\mathrm{p}0}$, which is $10\mathrm{bar}$ higher than the cap-side pressure, ${p}_{\mathrm{p}}$, opens valve C

_{2}, which remains open throughout the cylinder motion. In the end, the presence of valve V

_{p}does not establish the right opening sequence of the compensation valves; it, however, ends up creating a low-pressure zone at the cap-side when the cylinder starts moving, as seen in Figure 15b.

_{p}and V

_{a}, are placed at the cap- and rod-sides of the cylinder, so that a minimum pressure is created at the pump ports before the cylinder starts moving in either direction. The circuit layout is similar to the one shown in Figure 13, and, again, introduces inline flow resistances in the main circuit. Because of the similarities between this design and the previous design, we do not show the simulation results in this case.

## 4. Alternatives to Circuit with Pilot-Operated Check Valves

#### 4.1. Circuit Using Two-Position, Two-Way Directional Valves

_{1}and C

_{2}are adjusted to open at different cracking pressures, ${p}_{\mathrm{cr}1}$ and ${p}_{\mathrm{cr}2}$, respectively. The tank is also replaced with two constant pressure flow sources operating at pressures ${p}_{\mathrm{c}1}$ and ${p}_{\mathrm{c}2}$. The following conditions apply

_{1}and C

_{2}do not open. In this case, major problems are not expected to occur during the first and second quadrants of operation, because of the anti-cavitation valves C

_{01}and C

_{02}. However, third quadrant operation is impossible, since, in that case, valves C

_{01}and C

_{1}are simultaneously closed.

_{1}and C

_{2}, remain closed and the only reason why the cylinder extends smoothly is the opening of anti-cavitation valve C

_{02}, because of the lower pressure ${p}_{\mathrm{a}}<{p}_{0}$, at the cylinder rod-side (Figure 18b).

_{1}and C

_{2}open simultaneously (Figure 19a). The reason why the circuit behaves so erratically can be understood from the pressure curves in Figure 19b, where both cap- and rod-side pressures, ${p}_{\mathrm{p}}$ and ${p}_{\mathrm{a}}$, remain above the minimum activation pressure level for valves C

_{1}and C

_{2}($6\mathrm{bar}$) as soon as the external force, $F$, becomes positive. The cylinder, therefore, extends in a controlled manner by the pump, only during the second quadrant of operation.

_{1}and C

_{2}, with one single valve, so that when one side of the cylinder is connected to the compensation circuit, the other side is hydraulically disconnected. This is the idea behind the circuit in Figure 1d and will be analysed next.

#### 4.2. Circuit Using Three-Position, Three-Way Directional Valve with Different Activation Pressure Settings

**1**and

**2**in Figure 20. We then assume that when $\left({p}_{\mathrm{a}}-{p}_{\mathrm{p}}\right)=\left(k\alpha \right){p}_{\mathrm{cr}}$, with $k>1,$ the spool is completely displaced inside the valve case and the way between ports

**1**and

**2**is fully open. When $\left({p}_{\mathrm{a}}-{p}_{\mathrm{p}}\right)>\left(k\alpha \right){p}_{\mathrm{cr}}$, the spool does not move any longer and the valve behaves as a fixed orifice. With these considerations, we can write the following expressions for flows ${Q}_{\mathrm{c}1}$ and ${Q}_{\mathrm{c}2}$

- There is an ambiguity as to where the border between second and first quadrant divisions is located. Strangely enough, if we base such division as in Figure 3a, which has been the case in many publications [4,19,20], two different quadrant divisions arise given the same physical situation, as shown in Figure 21a and Figure 22a. This is simply because the spring stiffness and the flow resistance of the compensation valve directly influence the cap- and rod-side pressures. These pressures, on the other hand, affect the cylinder velocity and displacement, which is ultimately responsible for the external force (see Equations (6) and (7)).

_{p}and V

_{a}, thus introducing throttling losses.

_{p}and V

_{a}. As mentioned before, the circuits in Figure 20 and Figure 24 are equivalent when ${p}_{\mathrm{cr}}=0$ in Figure 20. Therefore, we have chosen to simulate the situation where the circuit in Figure 20 is used to drive a mass $m=0.5\mathrm{kg}$ using ${p}_{\mathrm{cr}}=1.0\times {10}^{-6}\text{}\mathrm{bar}$ (remember that we cannot use ${p}_{\mathrm{cr}}=0$, in Equations (28) and (29)).

## 5. New Operational Quadrant Division and the Solution for the Compensation Flow Shifting

**y**and

**z**are activated by the following logical equations:

**y**and

**z**of the directional valve, V, accordingly to the logical equations (30). Two anti-cavitation check valves, C, and two relief valves, R, connect both sides of the circuit to the tank for safety. An external compensation circuit, F, is connected to the compensation valve V.

_{1}and C

_{2}and a signal processing valve V. These valves comprise two distinct modules: a flow compensation module and a signal processing module. The modules operate independently, in such a way that one can design a very small signal processing module, while the compensation valves are chosen according to the cylinder size.

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Costa, G.K.; Sepehri, N. Four-Quadrant Analysis and System Design for Single-Rod Hydrostatic Actuators. J. Dyn. Syst. Meas. Control
**2019**, 141, 1–15. [Google Scholar] [CrossRef] - Stelson, K.A. Saving the World’s Energy with Fluid Power. In Proceedings of the 8th JFPS International Symposium on Fluid Power, Okinawa, Japan, 25–28 October 2011; pp. 1–7. [Google Scholar]
- Costa, G.K.; Sepehri, N. Hydrostatic Transmissions and Actuators—Operation, Modelling and Applications, 1st ed.; John Wiley & Sons: Chichester, UK, 2015. [Google Scholar]
- Quan, Z.; Quan, L.; Zhang, J. Review of energy efficient direct pump controlled cylinder electro-hydraulic technology. Renew. Sustain. Energy Rev.
**2014**, 35, 336–346. [Google Scholar] [CrossRef] - Jalayeri, E.; Imam, A.; Zeljko, T.; Sepehri, N. A throttle-less single-rod hydraulic cylinder positioning system: Design and experimental evaluation. Adv. Mech. Eng.
**2015**, 7, 1–14. [Google Scholar] [CrossRef] [Green Version] - Rahmfeld, R.; Ivantysynova, M. Energy Saving Hydraulic Actuators for Mobile Machines, 1st ed.; Bratislavian Fluid Power Symposium: Casta Pila, Slovakia, 1998; pp. 47–57. [Google Scholar]
- Imam, A.; Rafiq, M.; Jalayeri, E.; Sepehri, N. Design, Implementation and Evaluation of a Pump-Controlled Circuit for Single Rod Actuators. Actuators
**2017**, 6, 10. [Google Scholar] [CrossRef] [Green Version] - Caliskan, H.; Balkan, T.; Platin, E.B. A Complete Analysis and a Novel Solution for Instability in Pump Controlled Asymmetric Actuators. J. Dyn. Syst. Meas. Control
**2015**, 137, 091008. [Google Scholar] [CrossRef] - Wang, L.; Book, W.J.; Huggins, J.D. A Hydraulic Circuit for Single Rod Cylinder. J. Dyn. Syst. Meas. Control ASME
**2012**, 134, 011019. [Google Scholar] [CrossRef] [Green Version] - Imam, A.; Sepehri, N. Pump-Controlled Hydraulic Circuits for Operating a Differential Hydraulic Actuator. US Patent 2018/0266447, 20 September 2018. [Google Scholar]
- Altare, G.; Vacca, A. A design solution for efficient and compact electro-hydraulic actuators. Procedia Eng.
**2015**, 106, 8–16. [Google Scholar] [CrossRef] [Green Version] - Costa, G.K.; Sepehri, N. Logic-Controlled Flow Compensation Circuit for Operating Single-Rod Hydrostatic Actuators. International Patent WO/2019/051582, 21 March 2019. [Google Scholar]
- Costa, G.K.; Sepehri, N. A Critical Analysis of Valve-Compensated Hydrostatic Actuators: Qualitative Investigation. Actuators
**2019**, 8, 59. [Google Scholar] [CrossRef] [Green Version] - Stringer, J. Hydraulic System Analysis: An Introduction; John Wiley & Sons: New York, NY, USA, 1976. [Google Scholar]
- Yanada, H.; Khaing, W.H.; Tran, X.B. Effect of friction model on simulation of hydraulic actuator. In Proceedings of the 3rd International Conference on Design Engineering and Science, Pilsen, Czech Republic, 31 August–3 September 2014; pp. 175–180. [Google Scholar]
- Ren, G.; Esfandiari, M.; Song, J.; Sepehri, N. Position control of an electrohydrostatic actuator with tolerance to internal leakage. IEEE Trans. Control Syst. Technol.
**2016**, 24, 6. [Google Scholar] [CrossRef] - Parker, H. Hydraulic Valves Industrial Standard; Catalogue MSG11-3500/UK; Parker Hannifin Corporation: Warwick, UK, 2019. [Google Scholar]
- Costa, G.K.; Sepehri, N. A critical review of the existing models for direct operated hydraulic relief valves with the proposal of a new modelling approach. Int. J. Fluid Power
**2017**. [Google Scholar] [CrossRef] - Vukovic, M.; Sgro, S.; Murrenhoff, H. STEAM—A mobile hydraulic system with engine integration. In Proceedings of the ASME/BATH 2013 Symposium on Fluid Power & Motion Control FPMC2013, Sarasota, FL, USA, 6–9 October 2013. [Google Scholar]
- Heybroek, K. Saving Energy in Construction Machinery Using Displacement Control Hydraulics: Concept Realization and Validation. Licentiate Thesis, Linköping University, Linköping, Sweden, 2008. [Google Scholar]
- Gøytil, P.H.; Padovani, D.; Hansen, M.R. A Novel Solution for the Elimination of Mode Switching in Pump-Controlled Single-Rod Cylinders. Actuators
**2020**, 9, 20. [Google Scholar] [CrossRef] [Green Version] - Ren, G.; Costa, G.K.; Sepehri, N. Position control of an electro-hydrostatic asymmetric actuator operating in all quadrants. Mechatronics
**2020**, 67, 102344. [Google Scholar] [CrossRef]

**Figure 3.**Operational quadrant divisions: (

**a**) External force versus velocity; (

**b**) actuator force versus velocity.

**Figure 5.**First quadrant operation of the circuit shown in Figure 1a: (

**a**) operational diagram; (

**b**) forces acting on the piston and rod.

**Figure 7.**External force, F, cylinder velocity, v, and check valve flows, Q

_{c1}and Q

_{c2}, for $m=250\mathrm{kg}$.

**Figure 9.**External force, cylinder velocity and check valve flows for $m=250\mathrm{kg}$ and ${p}_{\mathrm{c}}=20\mathrm{bar}$.

**Figure 11.**Cylinder velocity and check valve flows for $m=0.5\mathrm{kg}$, ${p}_{\mathrm{c}1}=5\mathrm{bar}$ and ${p}_{\mathrm{c}2}=10\mathrm{bar}$.

**Figure 12.**External force, cylinder velocity and check valve flows for $m=250\mathrm{kg}$, ${p}_{\mathrm{c}1}=5\mathrm{bar}$ and ${p}_{\mathrm{c}2}=10\mathrm{bar}$.

**Figure 14.**(

**a**) Cylinder velocity and check valve flows; (

**b**) circuit pressures $\left(m=0.5\mathrm{kg}\right)$.

**Figure 15.**(

**a**) External force, cylinder velocity and check valve flows; (

**b**) circuit pressures ($m=250\mathrm{kg}$).

**Figure 18.**(

**a**) Cylinder velocity and compensation flows; (

**b**) cap- and rod-side pressures $\left(m=0.5\mathrm{kg}\right)$.

**Figure 19.**(

**a**) External force, cylinder velocity and compensation flows; (

**b**) cap- and rod-side pressures ($m=250\mathrm{kg})$.

**Figure 21.**(

**a**) Cylinder velocity and compensation flows; (

**b**) cap- and rod-side pressures ($m=0.5\mathrm{kg}$ and ${p}_{\mathrm{cr}}=3\mathrm{bar}$).

**Figure 22.**(

**a**) Cylinder velocity and compensation flows; (

**b**) cap- and rod-side pressures ($m=0.5\mathrm{kg}$ and ${p}_{\mathrm{cr}}=0.1\mathrm{bar}$).

**Figure 23.**(

**a**) External force, cylinder velocity and compensation flows (${p}_{\mathrm{cr}}=3\mathrm{bar}$); (

**b**) cap- and rod-side pressures (${p}_{\mathrm{cr}}=3\mathrm{bar}$); (

**c**) external force, cylinder velocity and compensation flows (${p}_{\mathrm{cr}}=0.1\mathrm{bar}$); (

**d**) cap- and rod-side pressures (${p}_{\mathrm{cr}}=0.1\mathrm{bar}$).

**Figure 25.**(

**a**) Cylinder velocity and compensation valve flows; (

**b**) cap- and rod-side pressures ($m=0.5\mathrm{kg}$ and ${p}_{\mathrm{cr}}=1.0\times {10}^{-6}\text{}\mathrm{bar}$).

**Figure 27.**(

**a**) External force, cylinder velocity and compensation flows; (

**b**) input and output powers at the cylinder and cap-side compensation flow ($m=0.5\mathrm{kg}$ and ${p}_{\mathrm{cr}}=1.0\times {10}^{-6}\text{}\mathrm{bar}$).

**Figure 29.**Expected circuit flows at each quadrant of operation. Pumping quadrants are shown in darker grey and motoring quadrants are shown in lighter grey.

**Figure 33.**(

**a**) Cylinder velocity and compensation flows; (

**b**) input and output powers ($m=0.5\mathrm{kg}$).

**Figure 34.**(

**a**) External force, cylinder velocity and check valve flows; (

**b**) input and output powers $\left(m=250\mathrm{kg}\right)$.

Parameter | Value |
---|---|

Nominal pump flow | ${Q}_{0}=6\mathrm{L}/\mathrm{min}$ (t in seconds) |

Effective bulk modulus [14] | ${\beta}_{\mathrm{E}}=1.7\times {10}^{9}\text{}\mathrm{N}/{\mathrm{m}}^{2}$ |

Piston and annulus areas of the cylinder | ${A}_{\mathrm{p}}=10{\text{}\mathrm{cm}}^{2}$ and ${A}_{\mathrm{a}}=5{\mathrm{cm}}^{2}$ |

Cylinder stroke | $L=0.5\mathrm{m}$ |

Inner conduit volume | $V=100{\mathrm{cm}}^{3}$ |

Friction coefficients [15,16] | ${F}_{\mathrm{C}}=78\mathrm{N}$ |

${F}_{\mathrm{S}}=250\mathrm{N}$ | |

${C}_{\mathrm{S}}=\left(1/145\right)\text{}\mathrm{m}/\mathrm{s}$ | |

${F}_{\mathrm{v}}=350\mathrm{Ns}/\mathrm{m}$ | |

Velocity threshold value [3] | ${v}_{\mathrm{th}}=1.0\times {10}^{-4}\text{}\mathrm{m}/\mathrm{s}$ |

Gravity acceleration | $g=9.8\mathrm{m}/{\mathrm{s}}^{2}$ |

Quadrant | Expected Valve Status |
---|---|

I | C_{1}—closed; C_{2}—open |

II | C_{1}—open; C_{2}—closed |

III | C_{1}—open; C_{2}—closed |

IV | C_{1}—closed; C_{2}—open |

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## Share and Cite

**MDPI and ACS Style**

Koury Costa, G.; Sepehri, N.
A Critical Analysis of Flow-Compensated Hydrostatic Single Rod Actuators: Simulation Study. *Actuators* **2020**, *9*, 58.
https://doi.org/10.3390/act9030058

**AMA Style**

Koury Costa G, Sepehri N.
A Critical Analysis of Flow-Compensated Hydrostatic Single Rod Actuators: Simulation Study. *Actuators*. 2020; 9(3):58.
https://doi.org/10.3390/act9030058

**Chicago/Turabian Style**

Koury Costa, Gustavo, and Nariman Sepehri.
2020. "A Critical Analysis of Flow-Compensated Hydrostatic Single Rod Actuators: Simulation Study" *Actuators* 9, no. 3: 58.
https://doi.org/10.3390/act9030058