# Simulation of Hydraulic Cylinder Cushioning

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Bond Graph Model

_{f}, effort sources S

_{e}, transformers between physics domains TF, capacitances C, resistances R, and inertia I. Finally, in the 0-junctions, all effort values are equal across the bonds and in the 1-junctions, all flow values are equal across the bonds.

- Resistive and capacitive effects are lumped wherever appropriate.
- There is no leakage between the piston chambers
- Internal Friction of the cylinder is not considered
- Fluid inertia is not considered.
- The tank pressure is assumed to be equal with the atmospheric pressure.
- Newtonian fluid is considered for the analysis.
- The oil temperature and hydraulic fluid viscosity are constant.

#### 2.1. Cylinder Rod

_{i}is the inertial force and F

_{m}is the weight of the displaced reduced mass to the cylinder rod, considered constant.

_{reduced}, for single degree of freedom systems, such as the studied one, is established according to the well-known Eksergian equation [21,22]:

_{limit}element is included representing the mechanical limits of the piston, where an “infinite” resistance is generated when the cylinder reaches its end of stroke. In order to avoid an unrealistic stiff response, the mechanical elasticity of the cylinder body, once reaching the end of the stroke, is represented by a C

_{limit}element.

#### 2.2. Excavator Arm Model

- The inertial effects of the hydraulic cylinder mass are considered negligible.
- The hydraulic cylinder is only subjected to forces in the same direction as its axial axis.
- The frictional forces on the seals and links are negligible.
- Links and mechanical fixing points are perfectly rigid.
- The fixing points with the cylinder, the center of rotation, and the supported masses are aligned on the same axis.
- Only the action of the cylinder on the shown arm is considered.

_{reduced}), resulting from the effect of the mass forces and the geometry of the mechanism.

_{i}is the different forces present in the system and L is the distance traveled.

_{cylinder}is the velocity of the cylinder rod relative to cylinder body, v

_{own}is the velocity of the excavator arm referred to its center of mass, v

_{load}is the velocity of the displaced load and g is the gravity acceleration. In Figure 3, the m

_{own}, m

_{load}, and angles τ and δ are graphically described. It should be noted that the angles τ and δ are determined essentially identical by geometry.

_{reduced}, r

_{own}, r

_{load}, v

_{reduced}, and θ angle are graphically described in Figure 3. Finally, Equation (7) reduces to:

_{c reduced}is the kinetic energy of the reduced mass and ${E}_{ci}$ is the kinetic energy of the remaining masses of the system.

_{disk}is the radius of the disk acting as a load on the experimental device.

_{own}, as well as the determination of its center of mass, is carried out from a 3D model in a CAD tool (free software FreeCAD

^{®}V0.13) [26,27], obtaining the following value:

## 3. Cushioning Model

#### 3.1. Design Parameters

_{0}as the position where the piston entirely closes the outlet port of the cushioning chamber, the characteristic parameters of the studied cushioning design are detailed in Figure 6 for 5 perimeter grooves (G1 to G5). The grooves have a rectangular section of width b

_{i}and depth h

_{i}, separated from X

_{0}a distance L

_{i}, for i = 1 to 5. D is the diameter of the outlet port. The radial assembly clearance between piston and cylinder body inner wall is defined as e.

#### 3.2. Cushioning Phases

_{port}

_{,}works from the coincidence of the piston with the outlet section of the port until its total occlusion. It is modeled from the classical sharp-edged orifice turbulent flow equation, where the flow is proportional to the square root of the pressure drop in the orifice Δp [29]:

_{d}is the port discharge coefficient and ρ the density of the fluid. The pressure drop is considered equal to cushioning chamber pressure due to outlet port is discharging to tank at atmospheric pressure, i.e., Δp ≈ P

_{cushioning}. The effective flow section S

_{port}changes as a circular segment.

_{annular}, is modeled as,

_{groove,}appears. This flow is also modelled with the sharp-edged orifice flow equation where

_{groove}

_{,}exists in both sides of the piston in front of the cylinder port. As detailed in next section, the grooves discharge coefficient C

_{d}are determined from the numerical results of CFD simulation.

## 4. Computational Fluids Dynamics

^{®}version 6 [30,31] software (managed by OpenCFD Ltd., Reading, UK), using the Semi-Implicit Method for Pressure-Linked Equations (SIMPLE) resolution algorithm [32] implemented in simpleFOAM solver. The SIMPLE solver solves the standard Navier-Stokes equations in steady state, neglecting the effect of gravity, for incompressible and Newtonian fluids. Laminar flow is imposed in all the experiments.

_{d}are calculated using Equation (16) below, obtained solving the Equation (15). This considers the groove section, S

_{groove}, existing on both sides of the piston in front of the cylinder port.

_{p}is the piston stroke (in centimeters) and Q

_{groove}is the flow rate through the groove in (in L/min). The coefficients a, b, and c, detailed in Table 1, are determined by least-squared best-fit for the different established experiments.

_{d}value of ±0.07. On the other hand, the mathematical extrapolation of the plane equation of the discharge coefficients, due to the implied simplification, turn in a maximum RMSE error of ±0.04.

## 5. Experimental Method

## 6. Results

#### 6.1. Cylinder Retraction

#### 6.2. Cylinder Extension

#### 6.3. Simulation Analysis

^{−6}and an absolute and relative integration error of 10

^{−6}.

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Homuth, K.C. Single Directional Sealing Piston Ring. U.S. Patent 4,207,800, 17 June 1980. [Google Scholar]
- Callies, R.E. Cushion Hydraulic Cylinder. U.S. Patent 6,186,043, 13 February 2001. [Google Scholar]
- Boecker, M. Hydraulic Cylinder. U.S. Patent US 7,171888, 6 February 2007. [Google Scholar]
- Algar, A.; Codina, E.; Freire, J. Experimental Study of 3D Movement in Cushioning of Hydraulic Cylinder. Energies
**2017**, 10, 746. [Google Scholar] [CrossRef] [Green Version] - Lie, T.; Chapple, P.J.; Tilley, D.G. Actuator cushion performance simulation and test results. In Proceedings of the PTMC2000 Workshop on Power Transmission and Motion Control, Bath, UK, 13 September 2000; pp. 187–198. [Google Scholar]
- Borghi, M.; Milani, M. Mechanical cushion design influence on cylinder dynamics. SAE Tech. Pap.
**2005**. [Google Scholar] [CrossRef] - Ding, F. Study on cushion process of high speed hydraulic cylinder. Iron Steel
**1998**, 33, 54–57. [Google Scholar] - Schwartz, C.; De Negri, V.J.; Climaco, J.V. Modeling and analysis of an auto-adjustable stroke end cushioning device for hydraulic cylinders. J. Braz. Soc. Mech. Sci. Eng.
**2005**, 27, 415–425. [Google Scholar] [CrossRef] [Green Version] - Lai, Q.; Liang, L.; Li, J.; Wu, S.; Liu, J. Modeling and Analysis on Cushion Characteristics of Fast and High-Flow-Rate Hydraulic Cylinder. Math. Probl. Eng.
**2016**, 2016, 2639480. [Google Scholar] [CrossRef] - Chen, X.; Chen, F.; Zhou, J.; Li, L.; Zhang, Y. Cushioning structure optimization of excavator arm cylinder. Autom. Constr.
**2015**, 53, 120–130. [Google Scholar] [CrossRef] - Kim, J.-H.; Kang, H.; Han, S.; Kim, Y. Motion Characteristics of Hydraulic Actuator with Various Cushion Shapes. KSFM J. Fluid Mach.
**2015**, 18, 48–53. [Google Scholar] [CrossRef] - Muvengei, M.; Kihiu, J. Bond Graph Modeling of Inter-Actuator Interactions in a Multi-Cylinder Hydraulic System. Int. J. Aerosp. Mech. Eng.
**2011**, 5, 147–156. [Google Scholar] - Athanasatos, P.; Costopoulos, T. An efficient modeling procedure of the dynamic behavior of high pressure hydraulic systems. In Proceedings of the 2nd International Conference on Experiments/Process/System Modelling/Simulation/Optimization, 2nd IC-EpsMsO, Athens, Greece, 4–7 July 2007. [Google Scholar]
- Romero, G.; Félez, J.; Martínez, M.L.; Del Vas, J.J. Simulation of the hydraulic circuit of a wheel loader by using the bond graph technique. In Proceedings of the 22nd European Conference on Modelling and Simulation, ECMS 2008, Nicosia, Cyprus, 3–6 June 2008; Volume 5. [Google Scholar]
- Arvani, F.; Rideout, G.; Krouglicof, N.; Butt, S. Bond Graph Modeling of a Hydraulic Vibration System: Simulation and Control. In Proceedings of the International Conference on Integrated Modeling and Analysis in Applied Control and Automation, IMAACA 2011, Rome, Italy, 12–14 September 2011; p. 6. [Google Scholar]
- Tripathi, J.P.; Ghoshal, S.K.; Dasgupta, K.; Das, J. Bond graph modelling of a hydraulic cylinder-actuated planar manipulator. J. Braz. Soc. Mech. Sci. Eng.
**2017**, 39, 4275–4287. [Google Scholar] [CrossRef] - Xiao, J.; Liu, Q.; Wang, G.; Ji, J. Theoretical and Experimental Analysis of the Hydraulic Actuator Used in the Active Reflector System. Math. Probl. Eng.
**2018**, 2018, 8503628. [Google Scholar] [CrossRef] - Afshari, H.H.; Zanj, A.; Novinzadeh, A.B. Simulation Modelling Practice and Theory Dynamic analysis of a nonlinear pressure regulator using bondgraph simulation technique. Simul. Model. Pract. Theory
**2010**, 18, 240–252. [Google Scholar] [CrossRef] - Gad, O. Bond Graph Modeling of a Two-Stage Pressure Relief Valve. J. Dyn. Syst. Meas. Control
**2013**, 135, 041001. [Google Scholar] [CrossRef] - Muvengei, M.; Kihiu, J. Bond Graph Modeling of Mechanical Dynamics of an Excavator for Hydraulic System Analysis and Design. Int. J. Mech. Ind. Aerosp. Eng.
**2009**, 3, 248–256. [Google Scholar] [CrossRef] - Cardona Foix, S.; Nebot, L.J.; Puig-Ortiz, J. Reduced inertial parameters in system of one degree of freedom obtained by Eksergian’s method. Mech. Sci.
**2017**, 8, 91–100. [Google Scholar] [CrossRef] [Green Version] - DrD. Mechanical Corner. Eksergian’s Equation of Motion for SDOF. Available online: https://mechanical-engg.com/blogs/entry/802-11-eksergians-equation-of-motion-for-sdof/ (accessed on 4 January 2021).
- Lee, S.J.; Chang, P.H. Modeling of a hydraulic excavator based on bond graph method and its parameter estimation. J. Mech. Sci. Technol.
**2012**, 26, 195–204. [Google Scholar] [CrossRef] - Koivo, A.J.; Thoma, M.; Kocaoglan, E.; Andrade-Cetto, J. Modeling and Control of Excavator Dynamics during Digging Operation. J. Aerosp. Eng.
**1996**, 9, 10–18. [Google Scholar] [CrossRef] - FreeCAD Tutorials. Available online: https://wiki.freecadweb.org/Tutorials/ (accessed on 4 January 2021).
- FreeCAD Website. Available online: https://www.freecadweb.org (accessed on 21 November 2020).
- Ramón Moliner, P. Cinemáticas de Máquinas; Universidad Politécnica de Barcelona, Escuela Técnica Superior de Ingenieros Industriales: Barcelona, Spain, 1970. [Google Scholar]
- Castilla, R.; Alemany, I.; Algar, A.; Roquet, P.; Codina, E. Pressure drop coefficients for cushioning system of hydraulic cylinder with 5 grooves eccentric piston: A Computational Fluid Dynamic simulation. Energies
**2017**, 10, 1704. [Google Scholar] [CrossRef] [Green Version] - Merritt, H.E. Hydraulic Control Systems; Wiley: Hoboken, NY, USA, 1991. [Google Scholar]
- OpenFoam CFD Website. Available online: https://www.openfoam.com (accessed on 21 November 2020).
- Maric, T.; Höpke, J.; Mooney, K. The OpenFOAM Technology Primer; Sourceflux UG: Duisburg, Germany, 2014. [Google Scholar]
- Moukalled, F.; Mangani, L. The Finite Volume Method in Computational Fluid Dynamics. An Advanced Introduction with OpenFoam® and Matlab®; Springer International Publishing: Cham, Switzerland, 2016. [Google Scholar]
- Milani, M. Designing hydraulic locking balancing grooves. Proc. Inst. Mech. Eng. Part I J. Syst. Control Eng.
**2001**, 215, 453–465. [Google Scholar] [CrossRef]

**Figure 3.**Geometry of the excavator’s arm and representation of the existing forces and masses. (

**a**) Lifting movement of the excavator arm (retraction hydraulic cylinder) and (

**b**) lowering movement of the excavator arm (extension hydraulic cylinder).

**Figure 7.**Radial displacement evolution during extension cushioning. Experimental registers from [4].

**Figure 9.**3D model of the cushioning chamber. (

**a**) Cylinder body, in section cut, and piston; (

**b**) Internal delimited volume.

**Figure 14.**Discharge coefficients C

_{d}for piston version 1. (

**a**) Simulation results and (

**b**) calculated plane.

**Figure 17.**Experimental records of velocity (

**left**) and cushioning pressure (

**right**) during retraction. 78 kg load. Groove stroke in front of the port as arrows.

**Figure 18.**Experimental records of velocity (

**left**), cushioning pressure (

**center**) and inlet pressure (

**right**) during retraction. Comparison with of version 1 and version 3 designs.

**Figure 19.**Experimental records of velocity (

**left**), cushioning pressure (

**center**) and inlet pressure (

**right**) during extension for design version 1. 78 kg load.

**Figure 20.**Experimental records of velocity (

**left**), cushioning pressure (

**center**) and inlet pressure (

**right**) during extension for design version 3. 78 kg load.

**Figure 21.**Retraction cushioning for experimental and simulated results. Velocity (

**top left**), cushioning pressure (

**top right**), inlet pressure (

**bottom left**) and radial gap (

**bottom right**).

**Figure 22.**Extension cushioning for experimental and simulated results. Velocity (

**top left**), cushioning pressure (

**top right**), inlet pressure (

**bottom left**) and radial gap (

**bottom right**).

Piston Design | Version 1 | ||||
---|---|---|---|---|---|

Groove | G1 | G2 | G3 | G4 | G5 |

Radial position | Centered | Centered | Intermediate | Attached | Attached |

a | 0.0045 | 0.0075 | 0.0082 | 0.0085 | 0.0107 |

b | 0 | 0.0044 | 0.0030 | 0.0061 | 0.0046 |

c | 0.2197 | −0.0100 | 0.0755 | 0 | 0 |

h_{1} | h_{2} | h_{3} | h_{4} | h_{5} | b_{1} | b_{2} | b_{3} | b_{4} | b_{5} | L_{1} | L_{2} | L_{3} | L_{4} | L_{5} | e | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Version 1 | 1.4 | 1.4 | 1 | 0.9 | 0.5 | 0.8 | 0.8 | 0.8 | 0.8 | 0.8 | 2 | 6.5 | 11 | 16 | 20 | 0.25 |

Version 3 | 3.0 | 3.0 | 1.4 | 0.9 | 0.5 | 0.8 | 0.8 | 0.8 | 0.8 | 0.8 | 2 | 6.5 | 11 | 16 | 20 | 0.25 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Algar, A.; Freire, J.; Castilla, R.; Codina, E.
Simulation of Hydraulic Cylinder Cushioning. *Sustainability* **2021**, *13*, 494.
https://doi.org/10.3390/su13020494

**AMA Style**

Algar A, Freire J, Castilla R, Codina E.
Simulation of Hydraulic Cylinder Cushioning. *Sustainability*. 2021; 13(2):494.
https://doi.org/10.3390/su13020494

**Chicago/Turabian Style**

Algar, Antonio, Javier Freire, Robert Castilla, and Esteban Codina.
2021. "Simulation of Hydraulic Cylinder Cushioning" *Sustainability* 13, no. 2: 494.
https://doi.org/10.3390/su13020494