# Identification and Validation of Linear Friction Models Using ANOVA and Stepwise Regression

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Experimental Setup and Data Collection

_{1}, J

_{2}). Friction work of the clutch can be adjusted through variable configurations of the fly wheels J

_{1}and by engaging inertia J

_{2}.

_{m}) positioned in the midplane of the respective steel plate. To determine the mass temperature of the steel plate, the signals of the thermocouples distributed around the circumference are averaged.

_{a}) and differential speed (Δn) during non-steady slip. The clutch is closed by applying the axial force. The following multiple slip phases are characterized by acceleration of the clutch to a maximum differential speed Δn and immediately after reaching this differential speed, it is decelerated again with the same gradient to the initial speed of zero. The slip phases are repeated for a defined number of times (in this study 5). After the last slip phase, the clutch is briefly released. A cooling phase follows to allow the clutch components to cool down. This is defined by a fixed differential speed of Δn = 20 rpm and an axial force of F

_{a}= 100 N, which is maintained until the middle steel plate reaches a steady temperature. The low axial force ensures distribution of the cooling oil in the grooves around the circumference of the clutch.

_{top}which represents the value of the CoF at maximum differential speed. Figure 5 shows an explanation for the measured value for two types of friction characteristics (course of CoF over sliding velocity) leading to equivalent values of µ

_{top}. We evaluate µ

_{top}of the last slip phase of each slip cycle. We use the mean average of µ

_{top}from the last five slip cycles of each load stage as an input parameter for our linear friction models. The values µ

_{top}are used as output values for the model.

_{top}) of each slip cycle of system D117/MP-C during the first run of the full-factorial test. Each block consists of 160 data samples, respectively slip cycles.

## 3. Methods

_{f}main effects and all their two-fold interactions according to [27] can be described as

^{2}describes the model quality and continuously decreases with the removal of factors. It can be interpreted as the percentage of variability that can be described by the model and is defined as

_{r}and the number of factors included in the model n

_{m}into account:

## 4. Results

#### 4.1. Application of ANOVA with Cross Validation for Model Derivation

^{2}and ${\mathrm{R}}_{\mathrm{adj}}^{2}$ has a smaller number of parameters. Therefore, every ANOVA analysis leads to a range, where the number of parameters is low enough to avoid overfitting and high enough to allow for an accurate representation of the characteristic value µ

_{top}. After validating the statistical prerequisites with the residual plots, the models in Table 7 are chosen as the most promising ones. Friction systems D88, MC-B and D117, MP-C are used to derive models. Friction system D88, MP-A was only used for validation of those models and was not used for model derivation itself. Model 1 and model 2 show values of R

^{2}and ${\mathrm{R}}_{\mathrm{adj}}^{2}$ greater 99.5%, model 3 shows R

^{2}= 97.4% and ${\mathrm{R}}_{\mathrm{adj}}^{2}$ = 96.7%.

^{2}and ${\mathrm{R}}_{\mathrm{adj}}^{2}$ after the factor of feeding oil flow rate is removed. This drop suggests that the remaining factors contribute more to the quality of the model.

#### 4.2. Application of Stepwise Regression with Cross Validation for Model Derivation

#### 4.3. Model Validation

^{−}) and the solid lines representing 3 N/mm² (p

^{+}). Although each model was derived for one specific friction system, all models react to the change in clutch pressure for all friction systems and can reproduce the level of CoF. Considering the behavior of CoF, the models perform differently in specific areas. In particular, the non-linear and asymmetric behavior of D117-MP-C L-204, caused by strong temperature dependence of CoF, is not reproduced by the linear models. Here, friction system D88-MP-A L-103 is used for validation only.

^{−}) and solid lines 90 °C (ϑ

^{+}). It can be observed that all models react to the change in oil temperature and can reproduce the level of the CoF.

## 5. Discussion

## 6. Conclusions

_{top}. Reasonable factors and corresponding factor levels are used for the experimental investigations.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**ZF/FZG KLP-260 test rig—schematic sketch according to [32].

**Figure 5.**Explanation of µ

_{top}for two types of friction characteristics with equivalent values of µ

_{top.}

**Figure 6.**Trend plot of full factorial design of system D117/MP-C—values per cycle of varied parameters (oil injection temperature, nominal feeding oil flow rate, clutch pressure, maximum differential speed) and the measured variable (µ

_{top}).

**Figure 7.**Representative example of behavior of coefficients R

^{2}and ${\mathrm{R}}_{\mathrm{adj}}^{2}$ and RMSE values for model and cross validation over number of parameters n

_{p}, D88 MC-B, L-201.

**Figure 9.**Comparison of measurement and simulation of CoF for load stage LS3 for three models fitted to D88, MC-B.

**Figure 10.**Comparison of measurement and simulation of CoF for load stage LS3 for three models fitted to D88, MC-B, restricted to ∆n > 1 rpm.

**Figure 11.**Comparison of measurement and simulation of CoF for load stage LS1 for three models fitted to D88, MC-B, restricted to ∆n > 1 rpm.

**Figure 12.**Comparison of measurement and simulation of CoF for load stages LS13 and LS15 with different pressures 1 N/mm² and 3 N/mm², restricted to ∆n > 1 rpm.

**Figure 13.**Comparison of measurement and simulation of CoF for load stage LS1 for three models fitted to D88, MC-B, restricted to ∆n > 1 rpm.

**Table 1.**Technical data of ZF/FZG KLP-260 test rig according to [33].

variable small fly wheels | J_{1} = 0.12 … 0.75 kgm² |

basic inertia | J_{2} = 1.0 kgm² |

plate diameters | d = 75 … 260 mm |

max. torque | T_{f,max} = 2000 Nm |

differential speed | Δn = 0 … 7000 rpm |

slip speed | Δn = 0 … 140 rpm (creep drive) Δn = 0 … 7000 rpm (main drive) |

torque in slip mode | T_{f,slip,max} = 2000 Nm (creep drive)T _{f,slip,max} = 60 Nm (main drive) |

max. axial force | F_{a,max} = 40 kN |

feeding oil temperature | ϑ_{oil} = 30 … 150 °C |

feeding oil flow rate | $\stackrel{.}{v}$_{oil} = 0 … 7 L/min |

feeding oil pressure | p_{oil} = 0 … 6 bar |

**Table 2.**Overview of measurement accuracy on the ZF/FZG KLP-260 test rig (confidence level 95 %) extended version according to [33].

Measured Variable | Uncertainty |
---|---|

axial force | +/− 1.3% |

torque | +/− 0.4% |

CoF | +/− 1.3% |

speed (main drive) | +/− 0.2% |

speed (creep drive) | +/− 0.9% |

thermocouple type K class 1 | +/− 1.8 K (DIN)/+/− 0.3 K (Estimated from calibration) |

feeding oil pressure | +/− 0.1 bar |

axial force | +/− 1.3% |

Lubricant | Kinematic Viscosity at 40 °C | Kinematic Viscosity at 100 °C |
---|---|---|

L-103 | 35 mm²/s | 7 mm²/s |

L-201 | 47 mm²/s | 9 mm²/s |

L-205 | 30 mm²/s | 6 mm²/s |

System | Mean Diameter | Friction Lining | Lubricant |
---|---|---|---|

D117 | 117 mm | MP-C | L-205 |

D88 | 88 mm | MP-A | L-103 |

D88 | 88 mm | MC-B | L-201 |

Name | p/N/mm² | ∆n/rpm | Number of Slip Phases |
---|---|---|---|

E1 | 0.75 | 25 | 5 |

E2 | 3.0 | 25 | 5 |

E3 | 1.5 | 25 | 5 |

E4 | 1–5 | 50 | 5 |

E5 | 0.75 | 50 | 5 |

E6 | 3.0 | 50 | 5 |

EC | 0 | 20 | 1 |

Symbol | A | B | C | D |
---|---|---|---|---|

Factor | Oil Injection Temperature | Feeding Oil Flow | Clutch Pressure | Max. Differential Speed |

Low Level (−) | 40 °C | 0.25 mm³/mm²s | 1 N/mm² | 25 rpm |

High Level (+) | 90 °C | 2 mm³/mm²s | 3 N/mm² | 100 rpm |

Name | Factor Level | |||

LS1 | + | + | + | + |

LS2 | + | + | + | − |

LS3 | + | + | − | + |

LS4 | + | + | − | − |

LS5 | + | − | + | + |

LS6 | + | − | + | − |

LS7 | + | − | − | + |

LS8 | + | − | − | − |

LS9 | − | + | + | + |

LS10 | − | + | + | − |

LS11 | − | + | − | + |

LS12 | − | + | − | − |

LS13 | − | − | + | + |

LS14 | − | − | + | − |

LS15 | − | − | − | + |

LS16 | − | − | − | − |

Friction System | Number of Parameters | Model |
---|---|---|

D88, MC-B | 7 | c_{0} + c_{A}·A+c_{c}·C+c_{D}·D+c_{E}·E+c_{AC}·AC+c_{AD}·AD (model 1) |

D88, MC-B | 8 | c_{0} + c_{A}·A + c_{c}·C + c_{D}·D + c_{E}·E + c_{AC}·AC + c_{CD}·CD + c_{DE}·DE (model 2) |

D117, MP-C | 7 | c_{0} + c_{A}·A + c_{c}·C + c_{D}·D + c_{E}·E + c_{AC}·AC + c_{DE}·DE (model 3) |

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**MDPI and ACS Style**

Strobl, P.; Schermer, E.; Groetsch, D.; Pointner-Gabriel, L.; Voelkel, K.; Pflaum, H.; Stahl, K.
Identification and Validation of Linear Friction Models Using ANOVA and Stepwise Regression. *Lubricants* **2022**, *10*, 286.
https://doi.org/10.3390/lubricants10110286

**AMA Style**

Strobl P, Schermer E, Groetsch D, Pointner-Gabriel L, Voelkel K, Pflaum H, Stahl K.
Identification and Validation of Linear Friction Models Using ANOVA and Stepwise Regression. *Lubricants*. 2022; 10(11):286.
https://doi.org/10.3390/lubricants10110286

**Chicago/Turabian Style**

Strobl, Patrick, Elias Schermer, Daniel Groetsch, Lukas Pointner-Gabriel, Katharina Voelkel, Hermann Pflaum, and Karsten Stahl.
2022. "Identification and Validation of Linear Friction Models Using ANOVA and Stepwise Regression" *Lubricants* 10, no. 11: 286.
https://doi.org/10.3390/lubricants10110286