# Polynomial Multiple Regression Analysis of the Lubrication Effectiveness of Deep Drawing Quality Steel Sheets by Eco-Friendly Vegetable Oils

## Abstract

**:**

## 1. Introduction

## 2. Material and Methods

#### 2.1. Material

#### 2.2. Strip Drawing Test

_{N}was exerted with a spring of the correct stiffness and set bolt. The normal force exerted on the specimen was obtained by suitable deflection of the spring. The relationship between the displacement and the force of the spring deflection in the range of 1 and 16 mm was determined using a MultiTest 10-i testing machine. The approximation of the force–displacement characteristics resulted in R

^{2}= 0.990. The deflection of the spring was realised by rotating the bolt head considering that the pitch of the thread was equal to 1.25 mm.

_{T}was recorded through the load cell of the measuring test machine and registered by testXpert

^{®}testing software. Friction tests were carried out at a constant speed of 10 mm/s [24] at a temperature of 25 °C.

_{N}is the normal (pressure) force and F

_{T}is the friction force.

_{N}. For each of these ranges, we obtained about 250–350 discrete values of the COF. The mean COF was determined using the formula [24]:

_{N}is the normal force of the countersamples, b is the width of specimen and E = 205,000 MPa is the Young’s modulus of the sheet material.

_{k}of the most commonly used vegetable lubricants varies between 27 and 40 mm

^{2}/s. Therefore, the most common oils with a kinematic viscosity in this range were selected for testing. The following vegetable oils were selected: palm, sunflower, cotton seed, soybean, linseed and coconut. Moreover, to check the effectiveness of the lubrication, a reference test in dry friction conditions was conducted.

^{−2}to be obtained, which was comparable to the stamping process conditions [45].

#### 2.3. Analysis of Variance

_{0}hypothesis is tested using the F statistic to see if the coefficient a

_{p}(the degree of the polynomial) at the highest power is zero. The H

_{0}hypothesis states that the observed sequence of observations x

_{1}, …, x

_{n}comes from the sample with density f

_{0}(x). If the test result is positive, the highest coefficient is eliminated by lowering the degree of the polynomial by one. The procedure is repeated successively for decreasing degrees of polynomials, until a negative answer is obtained [47].

## 3. Results and Discussion

#### 3.1. Effectiveness of Lubrication

_{l}:

_{d}is the coefficient of friction determined in dry friction conditions and μ

_{l}is the COF determined in lubricated conditions.

_{l}-value. These are second degree polynomials. Palm oil (Figure 4a), sunflower oil (Figure 4b) and cotton seed oil (Figure 4c) showed a similar value of lubrication efficiency in a range between 11 and 16% according to the nominal pressure. Soybean oil (Figure 4d) showed slightly less favourable lubricating properties. On the other hand, linseed oil (Figure 4e) and coconut oil (Figure 4f) had the lowest ability to reduce the value of the COF. Their lubrication efficiencies did not exceed 7% and were about 12% for the lowest and highest nominal pressure, respectively. In general, the lubrication efficiency increased with increasing nominal pressure.

#### 3.2. Analysis of Variance

^{2}− 0.0015C

^{2}

^{2}− 0.00006C

^{2}

^{2}of the regression model was equal to 0.9419 (Table 3). Due to the small difference between the predicted R

^{2}(0.9112) and the adjusted R

^{2}(0.9299), it can be concluded that the model is adequate. The signal-to-noise ratio parameter value for a realistic model should be greater than 4. The calculated value of this coefficient in the model was over 33.9, so the regression model is adequate.

_{i}is residual, k is number of parameters in the regression equation and MSE is the mean square error of the model.

_{i}measures the distance of a given observation from the mean value of the variable x:

_{i}from $\overline{\mathrm{x}}$.

## 4. Summary

- The following vegetable oils ensured the best efficiency of sheet metal lubrication in terms of nominal pressures between 2 and 12 MPa: palm, sunflower and cotton seed; these oils decreased the value of the COF by about 11–16% depending on the nominal pressure.
- Linseed and coconut oils had the most unfavourable lubrication properties, reducing the COF by about 7–12% depending on the nominal pressure.
- The small difference between the predicted R
^{2}and the adjusted R^{2}(0.0187) and the F-value of 78.41 indicated that the polynomial multiple regression model is adequate. - In the ANOVA model, the correlation of both density and viscosity was a significant factor. Load pressure was the most significant factor at a probability level p < 0.0001.
- In the whole range of pressures considered, the increase in the viscosity of the oil caused a reduction in the value of the COF. The effect of oil density on the COF value was similar.
- The most unfavourable friction conditions occurred when there was low density and low viscosity of the oils at the same time.
- At the lowest considered value of oil viscosity (η
_{k}= 27 mm^{2}/s), the increase in the density of the oil caused a slight increase in the COF. However, the greater the viscosity of the oil, the faster the COF value decreased with increasing density.

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 4.**Effectiveness of the lubrication of (

**a**) palm oil, (

**b**) sunflower, (

**c**) cotton seed oil, (

**d**) soybean oil, (

**e**) linseed oil and (

**f**) coconut oil.

**Figure 6.**SEM micrographs of the specimen surfaces tested at 10 MPa in lubricated conditions using (

**a**) cotton seed oil, (

**b**) linseed oil and (

**c**) coconut oil.

**Figure 7.**SEM micrographs of the specimen surfaces tested at 12 MPa in dry friction at different magnifications: (

**a**) ×370 and (

**b**) ×840.

**Figure 8.**(

**a**) The predicted vs. actual values of coefficient of friction and (

**b**) the normal probability plot of the externally studentised residuals.

**Figure 11.**Response surface plots presenting the interaction between the density and viscosity of oils affecting the COF at (

**a**) the minimum pressure considered (p = 2 MPa), (

**b**) the average pressure considered (p = 7 MPa) and (

**c**) the maximum pressure considered (p = 12 MPa).

**Figure 12.**Response surface plots presenting the interaction between the density of the oils and the nominal pressures affecting the COF at (

**a**) the minimum viscosity considered (η

_{k}= 27 mm

^{2}/s), (

**b**) the average viscosity considered (η

_{k}= 32.15 mm

^{2}/s) and (

**c**) the maximum viscosity considered (η

_{k}= 39.4 mm

^{2}/s).

**Figure 13.**Response surface plots presenting the interaction between the viscosity of the oils and the nominal pressures affecting the COF at (

**a**) the minimum density considered (ρ = 0.914 g/cm

^{3}), (

**b**) the average density considered (ρ = 0.9205 g/cm

^{3}) and (

**c**) the maximum density considered (ρ = 0.938 g/cm

^{3}).

Parameter | Density ρ, g/cm^{3} | Kinematic Viscosity ηk, mm^{2}/s | Nominal Load, MPa |
---|---|---|---|

Levels of variability | 0.914, 0.916, 0.918, 0.919, 0.938 | 27, 27.5, 29, 34, 36, 39.4 | 2, 4, 6, 8, 10,12 |

Source | SS | DOF | Mean Square | F-Value | p-Value | Meaning |
---|---|---|---|---|---|---|

Model | 0.0013 | 6 | 0.0002 | 78.41 | <0.0001 | significant |

A—Density | 7987 × 10^{−6} | 1 | 7987 × 10^{−6} | 2.86 | 0.1014 | |

B—Viscosity | 0.0001 | 1 | 0.0001 | 19.87 | 0.0001 | |

C—Pressure | 0.0007 | 1 | 0.0007 | 237.36 | <0.0001 | |

AB | 0.0000 | 1 | 0.0000 | 8.03 | 0.0083 | |

B^{2} | 0.0000 | 1 | 0.0000 | 6.90 | 0.0136 | |

C^{2} | 0.0000 | 1 | 0.0000 | 4.65 | 0.0394 | |

Residual | 0.0001 | 29 | 2.79 × 10^{−6} | |||

Correlation Total | 0.0014 | 53 |

Standard Deviation | 0.0017 |
R^{2} | 0.9419 |
---|---|---|---|

Mean | 0.2217 | Adjusted R^{2} | 0.9299 |

Coefficient of variance. % | 0.7536 | Predicted R^{2} | 0.9112 |

Adequacy precision | 33.9435 |

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**MDPI and ACS Style**

Trzepieciński, T.
Polynomial Multiple Regression Analysis of the Lubrication Effectiveness of Deep Drawing Quality Steel Sheets by Eco-Friendly Vegetable Oils. *Materials* **2022**, *15*, 1151.
https://doi.org/10.3390/ma15031151

**AMA Style**

Trzepieciński T.
Polynomial Multiple Regression Analysis of the Lubrication Effectiveness of Deep Drawing Quality Steel Sheets by Eco-Friendly Vegetable Oils. *Materials*. 2022; 15(3):1151.
https://doi.org/10.3390/ma15031151

**Chicago/Turabian Style**

Trzepieciński, Tomasz.
2022. "Polynomial Multiple Regression Analysis of the Lubrication Effectiveness of Deep Drawing Quality Steel Sheets by Eco-Friendly Vegetable Oils" *Materials* 15, no. 3: 1151.
https://doi.org/10.3390/ma15031151