# Numerical Simulation of a Slipper Model with Multi-Lands and Grooves for Hydraulic Piston Pumps and Motors in Mixed Lubrication

## Abstract

**:**

## 1. Introduction

## 2. Theoretical Model

_{s}and the eccentric load W were changed according to a rectangular waveform, as shown in Figure 3.

#### 2.1. Basic Equations

_{s}are the roughness parameters, and * denotes the expectation. The boundary conditions of ${\overline{p}}_{f}{}^{\ast}$ for the main land (subscript: mn), inner land (in), and outer land (out) are set by

#### 2.2. Calculation Procedure

^{4}–10

^{6}. The convergence criterion in terms of fluid pressure was 10

^{−5}. The periodic calculation was iteratively conducted until the solutions converged. The convergence criteria were set such that all differences in the center clearance, recess pressure, rotating angles around the x and y axes, and mean power loss fell below 10

^{−2}. Regarding the cavitation condition, the fluid pressure in gage was replaced with zero when the value became negative.

^{−1}, respectively, and the other parameters are set as $H/\sigma =1$, ${\sigma}_{1}/{\sigma}_{2}=1$, ${\mathsf{\Omega}}^{\prime}/\mathsf{\Omega}=1$, and $\omega /\mathsf{\Omega}=0$.

## 3. Results and Discussion

^{−1}.

## 4. Discussion

## 5. Conclusions

## Funding

## Conflicts of Interest

## Nomenclature

$a$ | inner radius ratio of main land, = ${R}_{1}/{R}_{2}$ |

${a}^{\prime}$ | outer radius ratio of main land, = ${R}_{2}{}^{\prime}/{R}_{2}$ |

${a}_{in1}$ | inner radius ratio of inner land, = ${R}_{in1}/{R}_{2}$ |

${a}_{in2}$ | outer radius ratio of inner land, = ${R}_{in2}/{R}_{2}$ |

${a}_{out1}$ | inner radius ratio of outer land, = ${R}_{out1}/{R}_{2}$ |

${a}_{out2}$ | outer radius ratio of outer land, = ${R}_{out2}/{R}_{2}$ |

${d}_{e}{}^{\ast}$ | separation |

$\tilde{E}\prime $$H$ | equivalent elastic modulus, = ${E}^{\prime}/\left({p}_{s0}{R}_{2}{}^{2}{S}_{0}\right)$ |

H | representative clearance |

${\tilde{H}}_{a}$ | hardness, = ${H}_{a}/\left({p}_{s0}{R}_{2}{}^{2}{S}_{0}\right)$ |

h | clearance, = $h/H$ |

${\overline{h}}_{c}$ | center thickness |

${h}_{T}{}^{\ast}$ | mean film thickness |

$\overline{I}$ | moment inertia, = ${\mathsf{\Omega}}^{2}HI/\left({p}_{s0}{R}_{2}{}^{4}{S}_{0}\right)$ |

$K$ | bulk modulus |

$\overline{L}$ | power loss, = $L/\left(\mathsf{\Omega}{p}_{s0}{R}_{2}{}^{3}{S}_{0}\right)$ |

$\overline{M}$ | moment, = $M/\left({p}_{s0}{R}_{2}{}^{3}{S}_{0}\right)$ |

$\overline{m}$ | mass, = ${\mathsf{\Omega}}^{2}Hm/\left({p}_{s0}{R}_{2}{}^{2}{S}_{0}\right)$ |

$\overline{p}$ | pressure, = $p/\left({p}_{s0}{S}_{0}\right)$ |

${\overline{p}}_{e}$ | ambient pressure, = ${p}_{e}/\left({p}_{s0}{S}_{0}\right)$ |

${\overline{p}}_{r}$ | recess pressure, = ${p}_{r}/\left({p}_{s0}{S}_{0}\right)$ |

${\overline{p}}_{s}$ | supply pressure, = ${p}_{s}/\left({p}_{s0}{S}_{0}\right)$ |

Q | flow rate, = $Q/\left(\mathsf{\Omega}{R}_{2}{}^{3}\right)$ |

${\overline{Q}}_{out}$ | leaked flow rate, = ${Q}_{out}/\left(\mathsf{\Omega}{R}_{2}{}^{3}\right)$ |

${R}_{0}$ | revolution radius |

${R}_{2}$ | representative radius |

$\overline{r},\theta ,\overline{z}$ | coordinates, = r/R_{2}, θ, z/H |

${r}_{w}$ | load eccentricity |

${S}_{0}$ | parameter, = $6\mu \mathsf{\Omega}{\left({R}_{2}/H\right)}^{2}/{p}_{s0}$ |

$\overline{T}$ | friction torque, = $T/\left({p}_{s0}{R}_{2}{}^{3}{S}_{0}\right)$ |

${\overline{V}}_{r}$ | recess volume, = $6\mu \mathsf{\Omega}{V}_{r}/\left({H}^{2}K{R}_{2}{}^{2}\right)$ |

$\overline{W}$ | load, = $W/\left({p}_{s0}{R}_{2}{}^{2}{S}_{0}\right)$ |

${w}_{p}{}^{\ast}$ | plasticity index |

$X,Y,Z$ | coordinates |

$x,y,z$ | coordinates |

$\overline{\alpha}$ | pad inclination angle, = $\alpha {R}_{2}/H$ |

$\beta $ | restrictor parameter, = $4{H}^{3}{l}_{c}/\left(3{r}_{c}{}^{4}\right)$ |

$\beta \prime $ | equivalent radius of asperity summit |

${\zeta}_{0}$ | hydrostatic balance ratio |

$\eta $ | asperity density |

$\overline{\lambda}$ | stiffness, = $\lambda H/\left({p}_{s0}{R}_{2}{}^{2}{S}_{0}\right)$ |

$\mu $ | viscosity |

$\sigma $ | surface roughness, = ${\left({\sigma}_{1}{}^{2}+{\sigma}_{2}{}^{2}\right)}^{1/2}$ |

${\sigma}^{\ast}$ | standard deviation of asperity summit height |

$\tau $ | time, = $\mathsf{\Omega}t$ |

$\varphi $ | pad azimuth |

$\mathsf{\Omega}$ | representative angular velocity |

${\mathsf{\Omega}}^{\prime}$ | disk angular velocity |

$\omega $ | pad angular velocity |

Subscripts: | |

a | asperity, contact |

c | center |

f | fluid |

gri | inner groove |

gro | outer groove |

in | inner land |

m | time-average |

max | maximum |

min | minimum |

mn | main land |

out | outer land |

r | recess |

0 | reference, high pressure period |

1 | inside |

2 | outside |

## Appendix A

## Appendix B

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**Figure 4.**The comparison of changes in the center clearance ${\overline{h}}_{c}$ and minimum clearance ${\overline{h}}_{\mathrm{min}}$.

**Figure 5.**The comparison of changes in the pad inclined angle $\overline{\alpha}$ and the pad azimuth ϕ.

**Figure 6.**The comparison of changes in the maximum contact pressure ${\overline{p}}_{a,\mathrm{max}}$ and leakage flow rate ${\overline{Q}}_{out}$.

**Figure 7.**The comparison of changes in the cavitating area ratio ${\overline{A}}_{cav}/{\overline{A}}_{0}$ and the contacting area ratio ${\overline{A}}_{cnt}/{\overline{A}}_{0}$.

**Figure 9.**The effects of supply pressure ${p}_{s}$ on normalized maximum contact pressure ${\overline{p}}_{a,\mathrm{max}}{}^{\ast}$ and the leakage flow rate ${\overline{Q}}_{out}{}^{\ast}$.

**Figure 10.**The effects of supply pressure ${p}_{s}$ on normalized mean power loss ${\overline{L}}_{m}{}^{\ast}$ and stiffness ${\overline{\lambda}}^{\ast}$.

**Figure 11.**The effects of supply pressure ${p}_{s}$ on the ratio of leakage power loss to total power loss ${\overline{L}}_{mQ}/{\overline{L}}_{m}$.

**Figure 13.**The effects of rotational speed $N$ on normalized maximum contact pressure ${\overline{p}}_{a,\mathrm{max}}{}^{\ast}$ and the leakage flow rate ${\overline{Q}}_{out}{}^{\ast}$.

**Figure 14.**The effects of rotational speed N on normalized mean power loss ${\overline{L}}_{m}{}^{\ast}$ and stiffness ${\overline{\lambda}}^{\ast}$.

**Figure 15.**The effects of rotational speed N on ratio of leakage power loss to total power loss ${\overline{L}}_{mQ}/{\overline{L}}_{m}$.

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

${a}_{in1}$ | 0.5 | |

${a}_{in2}$ | 0.6 | |

$a$ | 0.7 | |

${a}^{\prime}$ | 0.8 | |

${a}_{out1}$ | 0.9 | |

${a}_{out2}$ | 1 | |

$K$ | 1 | GPa |

$m$ | 100 | g |

${R}_{0}/{R}_{2}$ | 2.4 | |

${R}_{2}$ | 12.5 | mm |

${r}_{c}$ | 0.3 | mm |

${r}_{w}/{R}_{2}$ | 0.08 | |

${\zeta}_{0}$ | 1.1 | |

$\mu $ | 28 | mPa·s |

$\rho $ | 875 | kg/m^{3} |

$\sigma $ | 1 | μm |

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**MDPI and ACS Style**

Kazama, T. Numerical Simulation of a Slipper Model with Multi-Lands and Grooves for Hydraulic Piston Pumps and Motors in Mixed Lubrication. *Lubricants* **2019**, *7*, 55.
https://doi.org/10.3390/lubricants7070055

**AMA Style**

Kazama T. Numerical Simulation of a Slipper Model with Multi-Lands and Grooves for Hydraulic Piston Pumps and Motors in Mixed Lubrication. *Lubricants*. 2019; 7(7):55.
https://doi.org/10.3390/lubricants7070055

**Chicago/Turabian Style**

Kazama, Toshiharu. 2019. "Numerical Simulation of a Slipper Model with Multi-Lands and Grooves for Hydraulic Piston Pumps and Motors in Mixed Lubrication" *Lubricants* 7, no. 7: 55.
https://doi.org/10.3390/lubricants7070055