# Tailoring the Bore Surfaces of Water Hydraulic Axial Piston Machines to Piston Tilt and Deformation

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## Abstract

**:**

## 1. Introduction

## 2. Axial Piston Machines and Their Piston–Cylinder Lubricating Interfaces

## 3. Design Concept

## 4. State-of-the-Art in Surface Shaping at the Piston–Cylinder Interface

## 5. Interface Model

## 6. The TPGA

#### 6.1. Algorithm Overview

#### 6.2. Stage 1, Step 1.1: Skeleton Points

#### 6.3. Stage 1, Step 1.2: Interpolation Scheme

#### 6.3.1. Interpolation Scheme for Sections 1 and 4

- Set ${y}_{\mathrm{s}1\mathrm{r}2,\mathrm{mp}}\u03f6{m}_{2}=2{m}_{2,3}$, where ${m}_{2}$ is the slope of the circle at skeleton point 2 in the ${x}_{\mathrm{pr}}-{y}_{\mathrm{pr}}$ coordinate system and ${m}_{2,3}$ is the slope of the straight line passing through skeleton points 2 and 3 (also in the in the ${x}_{\mathrm{r}}-{y}_{\mathrm{r}}$ coordinate system). This is done because the interpolation in Section 2 is tangent to that of Section 1, and so the ${m}_{2}$ set by the Section 1 interpolation scheme cannot be so low as to prevent the Section 2 interpolation from reaching skeleton point 3. ${m}_{2}$ is also set to this value in an effort eliminate designs with high leakage across the interface, as a low slope at skeleton point 2 (in the ${x}_{\mathrm{r}}-{y}_{\mathrm{r}}$ coordinate system) increases the deviation of the profile from the ${x}_{\mathrm{pr}}$-axis over interpolation Section 2, which may raise the film thickness there.
- Set ${y}_{\mathrm{s}1\mathrm{r}2,\mathrm{mp}}\u03f6{m}_{1}>0$, where ${m}_{1}$ is the slope of the circle at skeleton point 1 in the ${x}_{\mathrm{pr}}-{y}_{\mathrm{pr}}$ coordinate system.
- Set ${y}_{\mathrm{s}1\mathrm{r}2,\mathrm{mp}}\u03f6{b}_{\mathrm{c}}\le 0$ (for ${b}_{c}$ in the ${x}_{\mathrm{pr}}-{y}_{\mathrm{pr}}$ coordinate system).

#### 6.3.2. Interpolation Scheme for Sections 2 and 3

#### 6.4. Stage 2, Step 2.2: Adjusting to Deformation

## 7. Case Study

#### 7.1. Simulation Results for Unit 1

#### 7.2. Experimental Results and Discussion

## 8. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

APMSPD | Axial piston machine of swash plate design |

DC | Displacement chamber |

HP | High pressure |

IDC | Inner dead center |

LP | Low pressure |

OC | Operating condition |

ODC | Outer dead center |

RKA | Rotating kit A |

RKB | Rotating kit B |

TPGA | Tailored profile generator algorithm |

## Nomenclature

Symbols | Description | Units |

$({a}_{\mathrm{C}},{b}_{\mathrm{C}})$ | Center point of the circular arc describing Sections 1 and 4 of the TPGA profile | m |

${C}_{\mathrm{A}}$ | Control point at the DC end of the piston-cylinder interface | N |

${C}_{\mathrm{B}}$ | Control point at the case end of the piston-cylinder interface | N |

${d}_{\mathrm{K}}$ | Piston diameter | m |

${d}_{\mathrm{B}}$ | Bore diameter | m |

F | Force | N |

${F}_{\mathrm{aK}}$ | Force on piston due to acceleration in axial direction | N |

${F}_{\mathrm{DK}}$ | Force on piston end due to DC pressure | N |

${F}_{\mathrm{SK}}$ | Reaction force from the swash plate to the slipper pushing on it | N |

${F}_{\mathrm{SKy}}$ | y-component of ${F}_{\mathrm{SK}}$ | N |

${F}_{\mathrm{TG}}$ | Force on piston due to slipper-swash plate friction | N |

${F}_{\mathrm{TK}}$ | Force on piston due to viscous friction in interface | N |

${F}_{\omega \mathrm{K}}$ | Force on piston due to centrifugal effect | N |

${F}_{\omega \mathrm{K},\mathrm{P}1}$ | Component of ${F}_{\omega \mathrm{K}}$ acting in ${P}_{1}$ | N |

h | Film thickness | m |

${h}_{\mathrm{c}}$ | Film thickness in current time step | m |

${h}_{\mathrm{cl}}$ | Piston-bore diametrical clearance | m |

${h}_{\mathrm{min}}$ | Film thickness below which the piston and bore surfaces are considered to engage in mixed or solid friction | m |

${h}_{\mathrm{p}}$ | Film thickness in previous time step | m |

L | Half the distance between the skeleton points bordering Section 1 (or Section 4) | m |

${l}_{\mathrm{B}}$ | Bushing length | m |

${l}_{\mathrm{F}}$ | Guide length | m |

${l}_{\mathrm{T}}$ | Trace length | m |

${m}_{1}$ | Slope of the TPGA profile at skeleton point 1 | - |

${m}_{1,2}$ | Slope of the straight line between skeleton points 1 and 2 | - |

${m}_{2}$ | Slope of the TPGA profile at skeleton point 2 | - |

${m}_{2,3}$ | Slope of the straight line between skeleton points 2 and 3 | - |

${m}_{4,5}$ | Slope of the straight line between skeleton points 4 and 5 | - |

n | Pump speed | rpm |

p | Pressure | Pa |

${p}_{\mathrm{LP}}$, ${p}_{\mathrm{HP}}$ | Pressure in low-pressure port, pressure in high-pressure port | Pa |

${p}_{\mathrm{in}}$, ${p}_{\mathrm{out}}$ | Inlet pressure, outlet pressure | Pa |

${Q}_{\mathrm{out}}$ | Flow out of the pump | L/min |

${R}_{\mathrm{C}}$ | Radius of the circular arc describing Sections 1 and 4 of the TPGA profile | m |

${\widehat{v}}_{\mathrm{b}}$, ${\widehat{v}}_{\mathrm{t}}$ | Velocity of bottom surface abutting the lubricating interface, velocity of top surface abutting the interface | m/s |

${y}_{\mathrm{s}1\mathrm{r}2}$ | ${y}_{\mathrm{r}}$-coord. of the apex of the circular arc describing Sections 1 and 4 of the TPGA profile | m |

${y}_{\mathrm{s}1\mathrm{r}2,\mathrm{mp}}$ | Max. value of ${y}_{\mathrm{s}1\mathrm{r}2}$ permitted by the geometric constraints specified in Section 6.3.1 | m |

${T}_{\mathrm{in}}$ | Pump inlet temperature | °C |

$\beta $ | Swash plate angle | rad |

${\delta}_{1}$, ${\delta}_{2}$ | Deviation of TPGA profile from flat at the DC end of the guide length, deviation at the case end of the guide length | m |

${\delta}_{1}$ | Height of TPGA profile crown between skeleton points 2,3, and 4 | m |

${\eta}_{\mathrm{v}}$, ${\eta}_{\mathrm{m}}$, ${\eta}_{\mathrm{t}}$ | Volumetric pump efficiency, mechanical pump efficiency, total pump efficiency | - |

${\gamma}_{1\mathrm{l}}$,${\gamma}_{1\mathrm{r}}$ | User input setting how pronounced the curvature between in TPGA Sections 1 and 4 is | - |

${\gamma}_{2\mathrm{l}}$,${\gamma}_{2\mathrm{r}}$ | User input setting how much the corner of TPGA Sections 2 and 3 is rounded off | - |

$\phi $ | Drive shaft angle | deg |

$\rho $ | Density | kg/${\mathrm{m}}^{3}$ |

$\mu $ | Dynamic viscosity | Pa·s |

$\tau $ | Torque transferred through axle | Nm |

## Appendix A

Measured Parameter | Sensor Accuracy |
---|---|

n | ±1.5 rpm |

${p}_{\mathrm{in}}$ | ±0.15 bar |

${p}_{\mathrm{out}}$ | ±0.15 bar |

$\tau $ | ±0.2 Nm |

${Q}_{\mathrm{out}}$ | ±40 L/h |

## Appendix B

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**Figure 1.**Axial piston machine of swash plate design. (

**a**) Components; (

**b**) DC pressure; (

**c**) Piston-cylinder interface.

**Figure 2.**Forces on piston (see in [1] for details).

**Figure 4.**Piston tilt (

**a**) and deformation (

**b**) over the first $\phi =90$° of the high-pressure stroke.

**Figure 18.**(

**a**) TPGA inputs; (

**b**) TPGA bore profile; and (

**c**–

**e**) simulation results for ${p}_{\mathrm{LP}}$ = 2.5 bar, ${p}_{\mathrm{HP}}$ = 120 bar, a pump speed of 700 rpm, and an inlet temperature of 20 °C.

Input | Definition |
---|---|

${l}_{\mathrm{F}}$ | Guide length |

${d}_{\mathrm{K}}$ | Piston diameter |

${d}_{\mathrm{B}}$ | Bushing bore diameter over the guide length (cylinder block bore diameter for bushing-less units) |

$HP$ | Nominal fluid pressure in HP port |

Skeleton Pt. | Interpolation |
---|---|

Inputs | Inputs |

${\delta}_{1}$ | ${\gamma}_{1\mathrm{l}}$ |

${\delta}_{2}$ | ${\gamma}_{1\mathrm{r}}$ |

${\delta}_{3}$ | ${\gamma}_{2\mathrm{l}}$ |

${\gamma}_{2\mathrm{r}}$ |

Skeleton Pt. | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|

${x}_{\mathrm{pr}}$ | 0 | $\frac{{\delta}_{3}-{\delta}_{1}}{{m}_{4,5}}$ | $\frac{1}{2}\left(\right)open="("\; close=")">\left(\right)open="("\; close=")">\frac{{\delta}_{2}-{\delta}_{1}}{{m}_{4,5}}$ | $\frac{{\delta}_{2}-{\delta}_{3}}{{m}_{4,5}}+{l}_{\mathrm{F}}$ | ${l}_{\mathrm{F}}$ |

${y}_{\mathrm{pr}}$ | $-{\delta}_{1}$ | $-{\delta}_{3}$ | 0 | $-{\delta}_{3}$ | $-{\delta}_{2}$ |

# | Constraint | Equation |
---|---|---|

1 | The circle center lies on the line y=L. | ${a}_{\mathrm{c}}=L$ |

2 | (0,0) is on the circle. | ${\left({R}_{\mathrm{c}}\right)}^{2}={\left({a}_{\mathrm{c}}\right)}^{2}+{\left({b}_{\mathrm{c}}\right)}^{2}$ |

3 | The point $(L,{y}_{\mathrm{s}1\mathrm{r}2})$ is on the circle. | ${\left({R}_{\mathrm{c}}\right)}^{2}={(L-{a}_{\mathrm{c}})}^{2}+{({y}_{\mathrm{s}1\mathrm{r}2}-{b}_{\mathrm{c}})}^{2}$ |

${a}_{\mathrm{c}}=L$ |

${b}_{\mathrm{c}}=\frac{{\left({y}_{\mathrm{s}1\mathrm{r}2}\right)}^{2}-{\left({a}_{\mathrm{c}}\right)}^{2}}{2{y}_{\mathrm{s}1\mathrm{r}2}}$ |

${R}_{\mathrm{c}}=\left(\right)open="|"\; close="|">{y}_{\mathrm{s}1\mathrm{r}2}-{b}_{\mathrm{c}}$ |

Shape | # | Geometric Constraint |
---|---|---|

Ellipse | 1 | Ellipse is tangent to Line A at Pt. ${s}_{2}{1}^{\prime}$. |

2 | Ellipse is tangent to Line B at Pt. ${s}_{2}{3}^{\prime}$. | |

3 | Pt. ${s}_{2}2$ is on the ellipse. | |

4 | In the ${x}_{\mathrm{e}}-{y}_{\mathrm{e}}$ coordinate system, Pt. ${s}_{2}2$ is the maximum point on the ellipse. | |

5 | The slope of the ellipse at Pt. ${s}_{2}2$ in the ${x}_{\mathrm{e}}-{y}_{\mathrm{e}}$ coordinate system, is zero. | |

Line A | 1 | Line A is tangent to the ellipse in Section 1 at Pt. ${s}_{2}{1}^{\prime}$. |

2 | Line A passes through Pt. ${s}_{2}1$. | |

Line B | 1 | Slope is zero. |

2 | Line B passes through Pt. ${s}_{2}3$. |

**Table 7.**Commercial unit: Danfoss APP 38-46 type pump, unit 180B3071 [32].

Feature | Description |
---|---|

Speed range | 700 rpm–1500 rpm |

Outlet pressure range | 10 barg–80 barg |

Flow range | 302 L/min–658 L/min |

**Table 8.**Key differences between the stock unit from Table 7, Unit 1, and Unit 2.

Feature | Stock Unit | Unit 1 | Unit 2 | |
---|---|---|---|---|

RKA | RKB | |||

Bore surface shaping | None | Profile I | None | Profile II |

Bushing length | ${l}_{\mathrm{b},\mathrm{s}}$ | ${l}_{\mathrm{b},1}<{l}_{\mathrm{b},\mathrm{s}}$ | ${l}_{\mathrm{b},2\mathrm{A}}={l}_{\mathrm{b},\mathrm{s}}$ | ${l}_{\mathrm{b},2\mathrm{B}}={l}_{\mathrm{b},1}$ |

Diametrical piston-bore clearance | ${h}_{\mathrm{cl},\mathrm{s}}$ | ${h}_{\mathrm{cl},1}<{h}_{\mathrm{cl},\mathrm{s}}$ | ${h}_{\mathrm{cl},2\mathrm{A}}={h}_{\mathrm{cl},\mathrm{s}}$ | ${h}_{\mathrm{cl},2\mathrm{B}}<{h}_{\mathrm{cl},\mathrm{s}}$ |

Bushing material | ∼ | Brass | Polymer | Polymer |

Piston grooves | Yes | No | Yes | Yes |

Swash plate angle | ${\beta}_{\mathrm{s}}$ | ${\beta}_{1}>{\beta}_{\mathrm{s}}$ | ${\beta}_{2\mathrm{A}}={\beta}_{\mathrm{s}}$ | ${\beta}_{2\mathrm{B}}={\beta}_{\mathrm{s}}$ |

**Table 9.**Operating conditions measured (${p}_{\mathrm{in}}$ = 2.5 bar, ${T}_{\mathrm{in}}$ = 20 °C).

OC | 1 | 2 | 3 | 4 | 5 | 6 |
---|---|---|---|---|---|---|

${p}_{\mathrm{out}}$ [bar] | 60 | 60 | 60 | 100 | 90 | 90 |

n [rpm] | 700 | 1100 | 1500 | 700 | 700 | 1100 |

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## Share and Cite

**MDPI and ACS Style**

Ernst, M.; Vacca, A.; Ivantysynova, M.; Enevoldsen, G.
Tailoring the Bore Surfaces of Water Hydraulic Axial Piston Machines to Piston Tilt and Deformation. *Energies* **2020**, *13*, 5997.
https://doi.org/10.3390/en13225997

**AMA Style**

Ernst M, Vacca A, Ivantysynova M, Enevoldsen G.
Tailoring the Bore Surfaces of Water Hydraulic Axial Piston Machines to Piston Tilt and Deformation. *Energies*. 2020; 13(22):5997.
https://doi.org/10.3390/en13225997

**Chicago/Turabian Style**

Ernst, Meike, Andrea Vacca, Monika Ivantysynova, and Georg Enevoldsen.
2020. "Tailoring the Bore Surfaces of Water Hydraulic Axial Piston Machines to Piston Tilt and Deformation" *Energies* 13, no. 22: 5997.
https://doi.org/10.3390/en13225997