# Multiscale Structural Mechanics of Rotary Shaft Seals: Numerical Studies and Visual Experiments

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Materials

#### 2.2. Radial Load Measurements

#### 2.3. Visual Experiments

#### 2.4. Friction Torque Measurements

#### 2.5. FE Modeling

## 3. Results

#### 3.1. FE Analysis

#### 3.2. Contact Analysis

#### 3.3. Contact Pressure

#### 3.4. Radial Load

#### 3.5. Friction Torque

#### 3.6. Tangential Displacement

## 4. Conclusions

- FE analyses are suitable for both macroscopic and microscopic investigations of the structural mechanics of rotary shaft seals. Real measured surface data can be directly integrated into the numerical model.
- Frustrated total internal reflection provided a qualitative analysis of the contact pressure distribution based on the contact interruption curve. This offers possible improvements with regard to a higher resolution of the contact interruption curve for the experimental detection of the microscopic effects in the contact pressure distribution.
- Particle image velocimetry (PIV) offers a possibility to quantify the tangential distortion of the sealing edge surfaces.
- Depending on the contact definition (threshold $k$), there were average differences of $35\%$ to $75\%$ between the measured and computed contact widths. This difference was also evident in the study of the contact pressure distribution. Possible reasons for this are tolerance deviations of the used materials and test rigs. There is potential for improvement here with regard to a completely user-independent contact analysis without the specification of a threshold $k$.
- The measured and computed radial loads were in the same range. Furthermore, numerical analyses showed that the radial load in operation (with distorted sealing edge surface) was higher than in mounted condition (with a compressed seal edge).
- The measured and computed tangential displacements of the sealing edges showed a high agreement for coefficients of friction in a range of $\mu =0.25\u20130.35$. There were no major differences between the ideally smooth sealing edge surface and the real surface measurement data.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

Symbols | |

$b$ | Contact width |

${C}_{10}$ | Neo-Hookean material parameter |

$d$ | Nominal diameter |

${d}_{\mathrm{o}}$ | Outer shaft diameter |

${F}_{\mathrm{M}}$ | Measured load |

${F}_{\mathrm{R}}$ | Radial load |

${F}_{\mathrm{S}}$ | Segment load |

${F}_{\mathrm{T}}$ | Tangential friction load |

$g$ | Gray values |

${g}_{\mathrm{t}\mathrm{h}}$ | Threshold gray value |

${G}_{\mathrm{B}\mathrm{r}\mathrm{i}\mathrm{n}\mathrm{k}}$ | Dimensionless number [23] |

${G}_{\mathrm{h}\mathrm{y}\mathrm{d}}$ | Dimensionless (Gümbel/Hersey) number [24,25] |

$k$ | Threshold for contact analysis |

$n$ | Shaft rotation speed |

$n$ | Refractive index |

${p}_{l}$ | Line load |

${R}_{\mathrm{z}}$ | Maximum height of the roughness profile |

${S}_{\mathrm{q}}$ | Root-mean-square roughness height |

$t$ | Roundness deviation |

${T}_{\mathrm{M}}$ | Measured friction torque |

$u$ | Tangential displacement |

$x$ | Circumferential direction |

$\dot{x}$ | Circumferential sliding velocity |

$y$ | Axial direction |

$z$ | Radial direction |

$\eta $ | Dynamic viscosity |

$\vartheta $ | Temperature |

$\lambda $ | Wavelength of the light |

$\mu $ | Coefficient of friction |

$\varphi $ | Friction parameter [21,22] |

$\omega $ | Angular velocity |

Abbreviations | |

DIN | German Institute for Standardization (ger.: Deutsche Institut für Normung e.V.) |

FEA | Finite element analysis |

FKM | Fluororubber |

FVA | Research Association for Drive Technology (ger.: Forschungsvereinigung Antriebstechnik e.V.) |

ISO | International standards organization |

PIV | Particle image velocimetry |

PMMA | Polymethyl methacrylate |

RSS | Rotary shaft seal |

VG | Viscosity grade |

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**Figure 3.**Representative surface sections selected out of all measurement fields: (

**a**) ${S}_{\mathrm{q}}=2.70\mathsf{\mu}\mathrm{m}$, (

**b**) ${S}_{\mathrm{q}}=2.98\mathsf{\mu}\mathrm{m}$, (

**c**) ${S}_{\mathrm{q}}=3.35\mathsf{\mu}\mathrm{m}$.

**Figure 4.**Schematic illustration of a radial load measurement device for the split-shaft measuring method.

**Figure 7.**Qualitative measurement of the contact pressure distribution in the sealing contact: (

**a**) Schematic representation of the qualitative determination of the pressure distribution; (

**b**) Image processing for pressure distribution analysis.

**Figure 9.**Rough sealing edge surfaces (

**a**) ${S}_{\mathrm{q}}=2.70\mathsf{\mu}\mathrm{m}$, (

**b**) ${S}_{\mathrm{q}}=2.98\mathsf{\mu}\mathrm{m}$, (

**c**) ${S}_{\mathrm{q}}=3.35\mathsf{\mu}\mathrm{m}$.

**Figure 10.**Distorted sealing edge surfaces: (

**a**) ${S}_{\mathrm{q}}=2.70\mathsf{\mu}\mathrm{m}$ (${S}_{\mathrm{q}}=0.30\mathsf{\mu}\mathrm{m}$), (

**b**) ${S}_{\mathrm{q}}=2.98\mathsf{\mu}\mathrm{m}$ (${S}_{\mathrm{q}}=0.27\mathsf{\mu}\mathrm{m}$) and (

**c**) ${S}_{\mathrm{q}}=3.35\mathsf{\mu}\mathrm{m}$ (${S}_{\mathrm{q}}=0.37\mathsf{\mu}\mathrm{m}$) root mean square height after deformation in brackets.

**Figure 11.**Contact pressure distribution: (

**a**) ${S}_{\mathrm{q}}=2.70\mathsf{\mu}\mathrm{m}$ (${S}_{\mathrm{q}}=0.30\mathsf{\mu}\mathrm{m}$), (

**b**) ${S}_{\mathrm{q}}=2.98\mathsf{\mu}\mathrm{m}$ (${S}_{\mathrm{q}}=0.27\mathsf{\mu}\mathrm{m}$) and (

**c**) ${S}_{\mathrm{q}}=3.35\mathsf{\mu}\mathrm{m}$ (${S}_{\mathrm{q}}=0.37\mathsf{\mu}\mathrm{m}$) root mean square height after deformation in brackets.

**Figure 13.**Computed percentage contact areas: (

**a**) ${S}_{\mathrm{q}}=2.70\mathsf{\mu}\mathrm{m}$ (${S}_{\mathrm{q}}=0.30\mathsf{\mu}\mathrm{m}$), (

**b**) ${S}_{\mathrm{q}}=2.98\mathsf{\mu}\mathrm{m}$ (${S}_{\mathrm{q}}=0.27\mathsf{\mu}\mathrm{m}$) and (

**c**) ${S}_{\mathrm{q}}=3.35\mathsf{\mu}\mathrm{m}$ (${S}_{\mathrm{q}}=0.37\mathsf{\mu}\mathrm{m}$) root mean square height after deformation in brackets.

**Figure 15.**Contact pressure distribution: (

**a**) Measured qualitative contact pressure distribution (contact interruption curve); (

**b**) Computed contact pressure distribution.

**Figure 17.**Gümbel curve (Gray area: Considered friction coefficient in the numerical studies; Dashed lines: Operating point for measuring the tangential displacement).

**Figure 18.**Tangential displacement: (

**a**) Measured tangential displacements; (

**b**) Comparison of measured and computed displacements.

Parameters | Value | Determination Method |
---|---|---|

Material parameter ${C}_{10}$ | $1.568\mathrm{M}\mathrm{P}\mathrm{a}$ | Material parameter determination [33] |

Shaft diameter $d$ | $80\mathrm{mm}$ | - |

Friction coefficient $\mu $ | $0.25\u20130.67$ | Literature [45,46,47,48] |

Segment load ${F}_{\mathrm{S}}$ | $5.6\times {10}^{-3}\mathrm{N}$ | Radial load measurements [36] |

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**MDPI and ACS Style**

Grün, J.; Gohs, M.; Bauer, F.
Multiscale Structural Mechanics of Rotary Shaft Seals: Numerical Studies and Visual Experiments. *Lubricants* **2023**, *11*, 234.
https://doi.org/10.3390/lubricants11060234

**AMA Style**

Grün J, Gohs M, Bauer F.
Multiscale Structural Mechanics of Rotary Shaft Seals: Numerical Studies and Visual Experiments. *Lubricants*. 2023; 11(6):234.
https://doi.org/10.3390/lubricants11060234

**Chicago/Turabian Style**

Grün, Jeremias, Marco Gohs, and Frank Bauer.
2023. "Multiscale Structural Mechanics of Rotary Shaft Seals: Numerical Studies and Visual Experiments" *Lubricants* 11, no. 6: 234.
https://doi.org/10.3390/lubricants11060234