# Virtual Prototyping of Axial Piston Machines: Numerical Method and Experimental Validation

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

#### 1.1. External Loads on the Rotating Group

_{DB}depends on the displacement chamber pressure p

_{Di}and surface area A

_{D}. Second, the spring force F

_{FB}pushes the cylinder block in the direction of the valve plate, on the z-axis. The spring force prevents the block from tipping when running at high speed and very low pressure. Third, the force due to the friction F

_{TB}between the piston and the cylinder bore in the piston/cylinder interface. Furthermore, the force due to the centripetal acceleration of the piston/slipper assembly, F

_{ωK}, acts in the radial direction of the cylinder block. The main force is due to the pressurized fluid in the displacement chamber, F

_{DK}pushes the bottom of the piston in the direction of the swashplate. The force due to the inertia of the piston/slipper assembly acts on the z-axis, F

_{aK}; and finally, the force due to the friction between the piston and the cylinder bore also acts in the z-axis, F

_{TK}. The total sum of these forces, F

_{DK}, F

_{aK}, and F

_{TK}, must be reacted by the swashplate. The forces related to the piston/slipper assembly are all transmitted to the cylinder block summed into a resultant side force, F

_{RK}, which is the same force represented for a single piston on as F

_{RBi}(not represented in the figure). The external forces and moments need to be balanced by the forces and moments generated by the fluid film pressure field. Refer to Ivantysyn and Ivantysynova [37].

#### 1.2. Pressure Module

_{SKi}), slipper/swash plate (Q

_{SGi}), and cylinder block/valve plate (Q

_{SBi}) interfaces.

- Swash plate moments
- Flow and pressure ripple
- Volumetric efficiency (due to internal leakage and compressibility)

#### 1.3. Thermo-Elastohydrodynamic Model for the Lubricating Interfaces

## 2. Virtual Prototyping Axial Piston Machines Methodology

- Fluid film thickness between the piston and the cylinder bore, the cylinder block and the valve plate, and the slipper and the swash plate.
- Pressure fields in the fluid film in the lubricating interfaces.
- Leakage flows in all three lubricating interfaces.
- Energy dissipation due to viscous flow in all three lubricating interfaces.
- Temperature distributions in the fluid film and main pump parts (cylinder block, piston, valve plate, slipper, swash plate, and end case).
- Surface deformations of the cylinder block, piston, valve plate, slipper, and swash plate due to pressure and thermal loading of these parts.

#### 2.1. Preliminary Design

_{K}, pitch diameter d

_{B}, piston diameter d

_{K}, etc. The combination of these parameters defines the geometrical displacement of the machine. The shaft diameter calculation must be done first since it will constrain the size of all the other remaining dimensions. An odd number of pistons is most commonly selected because it results in a smaller flow and torque ripple as Ivantysyn and Ivantysynova [29] have shown. The remaining variables are calculated in an iterative process. The outer and inner cylinder block diameters depend on the shaft, the cylinder block spring, and the pitch diameter which depends on the maximum piston diameter. The piston diameter should be in a range where no physical limitations could be violated due to side forces on the piston resulting in large stresses on the solid body. The displacement angle β is the angle formed between the swash plate running surface plane and the valve plate running surface plane. The swash plate angle is a critical parameter influencing the dimensions of the machine and its performance.

_{O}, shown in Figure 7. One of the existing proposed solutions is to change the geometry of the slipper by extending the distance from the center of the slipper to the running surface [39]. The starting geometrical dimensions for the lubricating interfaces (cylinder block/valve plate and slipper/swash plate and piston/cylinder) also need to be generated in this step within the preliminary design. Within this process, the three lubricating interfaces are assumed to have parallel gaps resulting in fixed fluid film heights. This simplification allows finding the first main dimensions for the lubricating interfaces by neglecting hydrodynamic and elastohydrodynamic effects, i.e., considering only hydrostatic forces created in the fluid film. Because this is a simplified assumption, which on one side allows the calculation of main dimensions of these interfaces using analytical expressions, correction factors, so-called balancing factors are introduced to make up for the missing hydrodynamic and elastohydrodynamic (EHD) effects being present in a real machine. The balance factors were found in an experimental trial and error process by manufacturers over the last five decades. Similarly, for the piston/cylinder interface the required clearance is defined assuming a centered position of the piston in the cylinder bore. More details about the preliminary design and the range of balance factors can be found in Ivantysyn and Ivantysynova [37].

#### 2.2. Virtual Prototyping

- Lowest energy dissipation in the given range of operating conditions.
- A stable fluid film with sufficient load carrying ability.
- Low flow and torque ripple.
- No cavitation or incomplete filling

## 3. Valve Plate Timing Optimization

- Optimize pressure orifice in the displacement chamber to avoid cavitation.
- High volumetric efficiency.
- Low flow ripple.
- Low control effort on the swash plate.
- Low moment pulsation (∆M
_{x}and ∆M_{y}).

_{x}, ∆M

_{y}, and the mean M

_{x}.

#### 3.1. Valve Plate Optimization Design Parameters

#### 3.2. Optimization Results Example

_{hp}with the color representing the ∆M

_{x}. It is shown that ∆Q

_{hp}is all over the plot concerning the volumetric efficiency; there is not a high correlation between the two. Figure 11 show a high correlation between the volumetric efficiency and the ∆M

_{x}. The lowest values for ∆M

_{x}are only possible by compromising the volumetric efficiency of the axial piston machine at the volumetric efficiency operating condition. Both these figures also depict the recommended design out of this optimization with a red filled circle. It can be observed that it only has about 10% volumetric efficiency at the volumetric efficiency operating condition, but it performs well for both the ∆M

_{x}and ∆Q

_{HP}. A different publication will be issued in all the intricacies of valve plate optimization since it is highly critical for the overall performance of axial piston machines. Compressibility and internal leakage losses which impact volumetric efficiency can make up to 30% or more of the overall power losses in an axial piston machine.

## 4. Virtual Prototyping for Solid Bodies

## 5. Virtual Prototyping for Lubricating Interfaces

- Fluid film thickness between the piston and the cylinder bore, the cylinder block and the valve plate, and the slipper and the swash plate.
- Pressure fields in the fluid film in the lubricating interfaces.
- Leakage flows in all three lubricating interfaces.
- Energy dissipation due to viscous flow in all three lubricating interfaces.
- Temperature distributions in the fluid film and main pump parts (cylinder block, piston, valve plate, slipper, swash plate, and end case).
- Surface deformations of the cylinder block, piston, valve plate, slipper, and swash plate due to pressure and thermal loading of these parts.

#### 5.1. Material Selection

#### 5.2. Operating Conditions

#### 5.3. Cylinder Block/Valve Plate Interface Design Variables within Virtual Prototyping Variables

_{gi}, inner port opening diameter d

_{oi}, pitch diameter d

_{B}, outer port opening diameter d

_{oo}, outer gap diameter d

_{go}, and kidney length l

_{kD}as shown in Figure 18a. Other design parameters are the cylinder block length (l

_{B}) and cylinder block channel length (l

_{CanalB}) which are shown in Figure 18b. The elastic deformation influences the performance of the interface therefore it might be necessary to return to the solid bodies design represented in Section 4. Additionally to the parameters shown above, the authors have done previous research [42] where the influence of micro-surfacing at the cylinder block/valve plate interface is detailed.

#### 5.4. Slipper/Swash Plate Interface Design Variables within Virtual Prototyping Variables

_{inG}and d

_{outG}shown in as well. Similarly, as in the previous interface, the elastic deformation of the slipper and swash plate will have an impact on the lubricating interface. Therefore, the slipper and swash plate geometrical shape and material selection may need to be modified in a design iteration going through the process detailed in Section 4.

#### 5.5. Piston/Cylinder Interface Design Variables within Virtual Prototyping Variables

_{K}and the diameter of the bore d

_{Z}are the most critical dimensions since these together define the clearance of the lubricating interface which impacts the fluid flow through the gap and viscous friction. The length of the piston l

_{K}and the length of the bore l

_{f}are important as well as they define the rest of the lubricating interface geometry. The piston/cylinder interfaces performance is influenced by but not limited to the variables described in this section. Therefore, the solid bodies may need to be redesigned as shown in Section 4.

## 6. Experimental Results and Comparison Against Simulations

#### 6.1. Test Rig Configuration

#### 6.2. Experimental Results

#### 6.3. Experimental Results and Simulation Comparison

## 7. Discussion

## 8. Conclusions

_{x}and ΔM

_{y}, fluid-borne noise source flow ripple ΔQ

_{HP}, and control effort of the axial piston machine’s swash plate (mean Mx).

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

Symbols | Description | Units |

h | Fluid film height | m |

u | Velocity on x-axis | m/s |

v | Velocity on y-axis | m/s |

p | Pressure | Bar |

t | Time | S |

w | Velocity on z-axis | m/s |

K | Bulk modulus | GPa |

V | Volume | m^{3} |

Q | Volumetric flow | m^{3}/s |

α | Discharge coefficient | - |

Ar | Minimum cross-section area | m^{2} |

µ | Dynamic viscosity | Pa·s |

φ | Shaft angle | ° |

ρ | Density | Kg/ m^{3} |

Subscripts | Description | |

i | Individual displacement chamber | - |

a | Sliding surface | - |

b | Fixed surface | - |

DC | Displacement chamber | - |

SK | Piston/cylinder interface | - |

SB | Cylinder block/valve plate interface | - |

SG | Slipper/swash plate interface | - |

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**Figure 11.**Valve plate optimization results Volumetric Efficiency vs ∆Q

_{hp}(

**a**) and Volumetric Efficiency vs ∆M

_{x}(

**b**).

**Figure 15.**Example of swash plate von Mises stress distribution on loaded swash plate (

**a**) and surface deformation of the sliding surface in the normal direction to the surface (

**b**).

**Figure 18.**Main dimensions impacting the cylinder block/valve plate interface. Cylinder block’s bottom view with sealing land dimensions (

**a**) and lateral cross section with main dimensions (

**b**).

**Figure 23.**Closed circuit prototype computational model. Cross-section lateral view (

**a**) and isometric view (

**b**) are shown.

**Figure 30.**Volumetric efficiency measured compared against simulated at n = 1000 rpm, T = 52 °C, and Δp = 50–400 bar.

**Figure 31.**Volumetric efficiency measured compared against simulated at n = 2000 rpm, T = 52 °C, and Δp = 50–400 bar.

**Figure 32.**Volumetric efficiency measured compared against simulated at n = 3000 rpm, T = 52 °C, and Δp = 50–400 bar.

**Figure 33.**Mechanical efficiency measured compared against simulated at n = 1000 rpm, T = 52 °C, and Δp = 50–400 bar.

**Figure 34.**Mechanical efficiency measured compared against simulated at n = 2000 rpm, T = 52 °C, and Δp = 50–400 bar.

**Figure 35.**Mechanical efficiency measured compared against simulated at n = 3000 rpm, T = 52 °C, and Δp = 50–400 bar.

**Figure 36.**Drain flow measured compared against simulated at n = 1000 rpm, T = 52 °C, and Δp = 50–400 bar.

**Figure 37.**Drain flow measured compared against simulated at n = 2000 rpm, T = 52 °C, and Δp = 50–400 bar.

**Figure 38.**Drain flow measured compared against simulated at n = 3000 rpm, T = 52 °C, and Δp = 50–400 bar.

Cost Functions |
---|

f_{1}($\overline{x}$) = Leakage (%) |

f_{2}($\overline{x}$) = ∆Q_{hp} (L/min) |

f_{3}($\overline{x}$) = ∆M_{x} (Nm) |

f_{4}($\overline{x}$) = ∆M_{y} (Nm) |

f_{5}($\overline{x}$) = $\overline{\mathrm{Mx}}$ (Nm) |

Description | Symbol | Unit |
---|---|---|

Young’s modulus | E | (Pa) |

Poisson’s ratio | ν | (-) |

Density | ρ | (kg/m^{3}) |

Thermal conductivity | λ | (W/mK) |

Coefficient of linear thermal expansion | α | (-) |

Operating Condition | Speed (rpm) | Δp (bar) | Displacement (%) |
---|---|---|---|

1 | max | max | max |

2 | max | max | min |

3 | max | min | min |

4 | max | min | max |

5 | min | max | max |

6 | min | max | min |

7 | min | min | min |

8 | min | min | max |

9 | min | max | moderate |

10 | moderate | max | max |

Description | Specification |
---|---|

Pressure differential | 50, 100, 200, 300, 400 (bar) |

Speed | 1000, 2000, 3000 (rpm) |

Displacement | 100 (%) |

Temperature | 42, 52, 72 (°C) |

ID | Description | Specification |
---|---|---|

1 | Electric drive | Max power: 225 Kw, Max torque 615 Nm @3500 rpm |

2, 3 | Staiger Mohilo torque cell | 0–500 Nm range, error ±0.2% of full scale |

4 | Closed circuit pump | 24cc, fixed displacement, max torque 160 Nm @Δp = 400 bar |

5 | Pressure transducer | WIKA S-10, 0–100 bar, 0.125% BFSL |

6, 8, 11 | Thermocouple | Omega K-type Thermocouple, 2.2 °C error limit |

7 | Pressure transducer | HYDAC HAD 4445, 0.5% BFSL |

9 | Flowmeter | VSE VS 10 Gear type, 1.2–250 L/min, 0.3% accuracy |

10 | Pressure transducer | WIKA S-10, 0–25 bar, 0.125% BFSL |

12 | Flowmeter | VSE VS 0.2 Gear type, 0.02–18 L/min, 0.3% accuracy |

13 | Pressure relief | Max flow 350 L/min |

14 | DAQ | NI cDAQ, NI 9213 |

15 | DAQ | NI cDAQ, NI 9201 |

© 2019 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**

Chacon, R.; Ivantysynova, M.
Virtual Prototyping of Axial Piston Machines: Numerical Method and Experimental Validation. *Energies* **2019**, *12*, 1674.
https://doi.org/10.3390/en12091674

**AMA Style**

Chacon R, Ivantysynova M.
Virtual Prototyping of Axial Piston Machines: Numerical Method and Experimental Validation. *Energies*. 2019; 12(9):1674.
https://doi.org/10.3390/en12091674

**Chicago/Turabian Style**

Chacon, Rene, and Monika Ivantysynova.
2019. "Virtual Prototyping of Axial Piston Machines: Numerical Method and Experimental Validation" *Energies* 12, no. 9: 1674.
https://doi.org/10.3390/en12091674