# A Digital Twin for Friction Prediction in Dynamic Rubber Applications with Surface Textures

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## Abstract

**:**

## 1. Introduction

#### 1.1. Surface Texturing in Tribological Applications

#### 1.2. Reduced Order Modelling

#### 1.3. Objectives

## 2. Materials and Methods

#### 2.1. Rubber Specimen Geometry and Surface Texture Parameters

#### 2.2. Test Rig Setup and Experimental Procedure for Determining the Coefficient of Friction

#### 2.3. Method of Measurement for Real Dimple Dimensions

#### 2.4. Software Development for Reduced Order Modelling

#### 2.5. Reduced Order Modelling Data Pre-Processing for Friction Reduction

#### 2.6. Statistical Analysis of Real Dimple Dimensions

## 3. Results

#### 3.1. Measurement Results of the Real Dimple Dimensions and Definition of Dimensionless Dimple Parameters

#### 3.2. Reduced Order Modelling on Friction Coefficient Data

#### 3.3. Reduced Order Modelling on Pre-Processed Data for Friction Variations

#### 3.4. Statistical Analysis Results of Real Dimple Dimensions

#### 3.5. Experimental Friction Measurement Results and ROM Friction Prediction Outcome

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

AI | Artificial Intelligence |

CFD | Cumulative Density Function |

DoE | Design of Experiment |

DT | Digital Twin |

FEM | Finite Element Method |

LST | Laser Surface Texturing |

ML | Machine Learning |

Probability Density Function | |

PGD | Proper Generalized Decomposition |

POD | Proper Orthogonal Decomposition |

ROM | Reduced Order Modelling |

SEHL | Soft Elasto-Hydrodynamic Lubrication |

SVD | Singular Value Decomposition |

TDM | Texturing During Moulding |

TRD | Tensor Rank Decomposition |

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**Figure 1.**(

**a**) Geometry of the rubber specimens including the most relevant dimensions, (

**b**) picture of the rubber specimen with focus on the dimple texture and (

**c**) positioning of the dimples in relation to the relative velocity vector.

**Figure 2.**(

**a**) Pin-on-disc tribometer design, (

**b**) picture of the tribometer with focus on the rubber specimen and the rotating counter surface, and (

**c**) contact conditions between the rubber sample and the counter surface.

**Figure 3.**(

**a**) Assembly of the finite element method (FEM). The coloured spherical contact area of the rubber specimen is brought into contact with the grey flat counter surface. (

**b**) Parabolic contact pressure distribution ${p}_{c}$ as a function of the contact diameter ${d}_{c}$.

**Figure 4.**(

**a**) Microscope image of rubber specimen 6, considering a measured area of 4 × 4 mm and (

**b**) related height profile h as function of the measuring length l, considering the indicated line scan trough four adjacent dimples.

**Figure 5.**(

**a**) Red marked nominal circular contact area ${A}_{nominal}$ between an untextured rubber specimen and the counter surface and (

**b**) contact area of a textured sample. The textured area ${A}_{textured}$ is indicated by the black circular dimples, the untextured area that is in direct contact with the counter surface is coloured in blue.

**Figure 6.**ROM prediction for (

**a**) untextured and (

**b**) textured friction coefficient data. The blue dots are obtained by evaluating the ROM with the experimental data, the dashed black line is the linear regression that fits the data, while the solid red line represents the ideal ROM result, where the ROM evaluation on each experimental friction value returns the same value.

**Figure 7.**2-ROM-models prediction for combined untextured and textured friction data, according to Equation (4). The blue dots are obtained by evaluating the combined 2-ROM predictions with the combined experimental data (Equation (4)), the dashed black line is the linear regression that fits the data, while the solid red line represents the ideal prediction result, where the predicted values perfectly match the experimental ones.

**Figure 8.**Textured data ROM’s first term. The one-dimensional functions, for each variable are shown separately in the plots, according to Equation (1), labelled with letters from (

**a**–

**e**).

**Figure 12.**(

**a**) ROM prediction with statistical noise introduction on depth and (

**b**) on diameter and distance. The blue dots are obtained by evaluating the ROM on the statistically expanded dataset, the solid red line represents the ideal ROM result and the error bars show the friction variations linked to (

**a**) depth or (

**b**) diameter and distance deviations from nominal values.

**Figure 13.**(

**a**) ROM prediction with statistical noise introduction on depth and (

**b**) on diameter and distance, with centred mean on nominal value and $\sigma $ = 0.01. The blue dots are obtained by evaluating the ROM on the statistically expanded dataset, the solid red line represents the ideal ROM result and the error bars show the friction variations linked to (

**a**) depth or (

**b**) diameter and distance deviations from nominal values.

**Figure 14.**(

**a**) Friction coefficient $\mu $ as function of the relative velocity ${v}_{r}$ for the eight different rubber specimens and (

**b**) the corresponding friction variation $\Delta \mu $ in relation to the untextured rubber sample 8, for contact pressures of ${p}_{c,max}$ = 0.5 MPa.

**Figure 15.**(

**a**) Friction coefficient $\mu $ as function of the relative velocity ${v}_{r}$ for the eight different rubber specimens and (

**b**) the corresponding friction variation $\Delta \mu $ in relation to the untextured rubber sample 8, for contact pressures of ${p}_{c,max}$ = 0.7 MPa.

**Figure 16.**(

**a**) Friction coefficient $\mu $ as function of the relative velocity ${v}_{r}$ for the eight different rubber specimens and (

**b**) the corresponding friction variation $\Delta \mu $ in relation to the untextured rubber sample 8, for contact pressures of ${p}_{c,max}$ = 0.9 MPa.

Sample i | Dimple Diameter [$\mathsf{\mu}$m] | Dimple Distance [$\mathsf{\mu}$m] | Dimple Depth [$\mathsf{\mu}$m] |
---|---|---|---|

1 | 100 | 300 | 10 |

2 | 200 | 100 | 10 |

3 | 200 | 200 | 10 |

4 | 300 | 100 | 30 |

5 | 300 | 200 | 20 |

6 | 300 | 200 | 30 |

7 | 300 | 300 | 20 |

8 | - | - | - |

**Table 2.**Operating parameters that are examined during the test procedure. (

**a**) Rotational speeds n of the servomotor and corresponding relative velocities ${v}_{r}$ between the rubber sample and the counter surface (Table 2), (

**b**) together with the normal force ${F}_{N}$, the related maximum contact pressure ${p}_{c,max}$, the contact diameter ${d}_{c}$ between the rubber sample and the counter surface, as well as the nominal contact area ${A}_{nominal}$ (Table 2).

a | |||
---|---|---|---|

Rotational Speedn[min${}^{-\mathbf{1}}$] | Relative Velocity${\mathit{v}}_{\mathit{r}}$ [mm/s] | ||

0.6 | 6 | ||

1.2 | 12 | ||

1.8 | 19 | ||

3.0 | 31 | ||

6.0 | 63 | ||

12.0 | 126 | ||

18.0 | 188 | ||

24.0 | 251 | ||

b | |||

Normal Force${\mathit{F}}_{\mathit{N}}$ [N] | Max Contact Pressure${\mathit{p}}_{\mathit{c},\mathit{max}}$ [MPa] | Contact Diameter${\mathit{d}}_{\mathit{c}}$ [mm] | Nominal Contact Area${\mathit{A}}_{\mathit{nominal}}$ [mm${}^{\mathbf{2}}$] |

3.9 | 0.5 | 4.2 | 13.8 |

7.9 | 0.7 | 5.0 | 19.6 |

13.3 | 0.9 | 5.8 | 26.4 |

**Table 3.**Real dimple dimensions, indicated by diameter, distance, and depth. The aspect ratio is the quotient of dimple depth and diameter. The textured area percentage is determined from the quotient of the textured area ${A}_{textured}$ and the nominal contact area ${A}_{nominal}$.

Sample i | Dimple Diameter [$\mathsf{\mu}$m] | Dimple Distance [$\mathsf{\mu}$m] | Dimple Depth [$\mathsf{\mu}$m] | Aspect Ratio | Textured Area [%] |
---|---|---|---|---|---|

1 | 135 | 258 | 16 | 0.11 | 9 |

2 | 242 | 100 | 11 | 0.05 | 39 |

3 | 241 | 165 | 10 | 0.04 | 28 |

4 | 337 | 66 | 35 | 0.10 | 55 |

5 | 330 | 170 | 22 | 0.06 | 34 |

6 | 346 | 153 | 35 | 0.10 | 37 |

7 | 336 | 252 | 20 | 0.06 | 25 |

8 | - | - | - | - | - |

**Table 4.**(

**a**) Untextured and (

**b**) Textured ROM ${R}^{2}$ for Train and Test datasets when the k-fold cross validation method is applied.

a | b | ||||
---|---|---|---|---|---|

Untextured ROM${\mathit{R}}^{\mathbf{2}}$ | Textured ROM${\mathit{R}}^{\mathbf{2}}$ | ||||

Train | Test | Train | Test | ||

1 | 1.000 | 0.895 | 1 | 1.000 | 0.914 |

2 | 1.000 | 0.968 | 2 | 1.000 | 0.872 |

3 | 1.000 | 0.988 | 3 | 1.000 | 0.882 |

4 | 1.000 | 0.974 | 4 | 1.000 | 0.943 |

5 | 1.000 | 0.988 | 5 | 1.000 | 0.923 |

6 | 1.000 | 0.903 | 6 | 1.000 | 0.896 |

7 | 1.000 | 0.871 | 7 | 1.000 | 0.963 |

8 | 1.000 | 0.994 | 8 | 1.000 | 0.912 |

9 | 1.000 | 0.915 | 9 | 1.000 | 0.953 |

10 | 1.000 | 0.982 | 10 | 1.000 | 0.938 |

avg | 1.000 | 0.948 | avg | 1.000 | 0.919 |

**Table 5.**Dimple dimensions predicted by the ROM, which further reduce friction based on the use cases.

Use Case Num. | Relative Velocity ${\mathit{v}}_{\mathit{r}}$ [mm/s] | Max Contact Pressure ${\mathit{p}}_{\mathit{c},\mathit{max}}$ [MPa] | Dimple Diameter [$\mathsf{\mu}$m] | Dimple Distance [$\mathsf{\mu}$m] | Dimple Depth [$\mathsf{\mu}$m] | Aspect Ratio | Textured Area [%] | Predicted Friction Reduction [%] |
---|---|---|---|---|---|---|---|---|

1 | 251 | 0.5 | 270 | 100 | 10 | 0.04 | 42 | 63 |

2 | 31 | 0.7 | 300 | 186 | 11 | 0.04 | 30 | 81 |

3 | 6 | 0.9 | 274 | 111 | 11 | 0.04 | 39 | 72 |

4 | 100 | 0.6 | 300 | 140 | 11 | 0.04 | 36 | 79 |

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## Share and Cite

**MDPI and ACS Style**

Zambrano, V.; Brase, M.; Hernández-Gascón, B.; Wangenheim, M.; Gracia, L.A.; Viejo, I.; Izquierdo, S.; Valdés, J.R.
A Digital Twin for Friction Prediction in Dynamic Rubber Applications with Surface Textures. *Lubricants* **2021**, *9*, 57.
https://doi.org/10.3390/lubricants9050057

**AMA Style**

Zambrano V, Brase M, Hernández-Gascón B, Wangenheim M, Gracia LA, Viejo I, Izquierdo S, Valdés JR.
A Digital Twin for Friction Prediction in Dynamic Rubber Applications with Surface Textures. *Lubricants*. 2021; 9(5):57.
https://doi.org/10.3390/lubricants9050057

**Chicago/Turabian Style**

Zambrano, Valentina, Markus Brase, Belén Hernández-Gascón, Matthias Wangenheim, Leticia A. Gracia, Ismael Viejo, Salvador Izquierdo, and José Ramón Valdés.
2021. "A Digital Twin for Friction Prediction in Dynamic Rubber Applications with Surface Textures" *Lubricants* 9, no. 5: 57.
https://doi.org/10.3390/lubricants9050057