# Parameter Identification of Cutting Forces in Crankshaft Grinding Using Artificial Neural Networks

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^{2}

^{3}

^{4}

^{5}

^{6}

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## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. Materials and Methods

#### 3.1. General Formulation of the Problem

_{z}of the cutting force [31]:

_{t}—ultimate strength of workpiece material at a high temperature (about 600 °C), kgf/mm

^{2}; H—sonic index of the grinding wheel; Z—grinding wheel grit, μm; V

_{p}—infeed speed, mm/min; S—the peripheral speed of workpiece rotation, m/min; S

_{pr}—the longitudinal speed of grinding wheel dressing, mm/min; t

_{pr}—dressing depth, mm; B—grinding width, mm.

_{k}(k = 1, 2,…, m), defined for specific technological conditions of production. However, there is no general method for determining these parameters.

- creating a method for conducting a virtual experiment to determine the cutting force depending on an array of m parameters of formula (1):
- creating a table of input data for the variation range of parameters that affect the cutting force;
- generating an array of experimental data as a sample of a sufficiently large number of n random arrays of input parameters:
- creating a subroutine for determining the total number of each input parameter;
- generating a set of n combinations of values of m input parameters with a predetermined relative error δ;
- calculation of n values of cutting forces as a result of a virtual experiment;
- determining the maximum values of each input parameter and the cutting force ${P}_{z}^{max}$;

- using artificial intelligence tools to identify parameters of a mathematical model based on experimental data:
- normalization of input and output parameters;
- creating an artificial neural network and configuring its parameters;
- training an artificial neural network based on an array of normalized experimental data;
- determining the accuracy of estimating the cutting-force value for an arbitrary set of input parameters;

- creating a reliable generalized mathematical model for estimating (m + 1) parameters that determine the cutting force using multi-parameter quasi-linear regression analysis:
- creating a matrix relation for determining the cutting force based on experimental data;
- formation of the stiffness matrix and the column vector of external influence:
- formation of column sub-vectors of external influence and local stiffness parameters;
- formation of the stiffness submatrix;
- globalization of submatrix and column sub-vectors to a common stiffness matrix;

- using the inverse matrix method to evaluate (m + 1) unknown parameters;
- forming a ratio for calculating the cutting force based on a specific coefficient and indices of power, and comparing the obtained dependence with the empirical formula (1);
- determination of relative errors in determining unknown coefficients of the regression model.

#### 3.2. Virtual Experiment on the Cutting Force

_{1}= σ

_{t}, x

_{2}= H, x

_{3}= V

_{p}, x

_{4}= Z, x

_{5}= S, x

_{6}= S

_{pr}, x

_{7}= t

_{pr}, x

_{8}= B.

_{z}, determined as a result of conducting the i-th experiment.

## 4. Results

#### 4.1. Application of Artificial Intelligence Systems

_{k}are shown in Table 1.

^{®}software. The architecture of the ANN is shown in Figure 1 a.

^{6}; target error—1 × 10

^{−5}; initialization method of the threshold—random; initialization method of weighting factor—random; analysis update interval—500 cycles.

^{−3}; the average error per output per dataset—1.6 × 10

^{−5}. Thus, the training accuracy of an artificial neural network is relatively high.

#### 4.2. Multi-Parameter Quasi-Linear Regression Procedure

_{k}using the following dependence:

_{0}, as well as the submatrix [D] and the column sub-vectors {S} and {Y}, whose elements are defined by the following formulas:

^{4}, the calculation results, including the estimation procedure’s errors, are summarized in Table 3.

_{4}, β

_{5}, β

_{6}, β

_{6}confirm the inversely proportional dependence of the cutting force P

_{z}on the parameters Z, S, S

_{pr}, t

_{pr}in formula (1).

_{k}is in the range of 0.8–1.5%. As a result, the total relative error in determining the cutting force does not exceed 5.0%.

_{p}significantly impact the evaluated data (on average, 1.9% and 1.6%, respectively).

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Artificial neural network architecture (

**a**) and convergence of the regression procedure (

**b**).

Parameter | σ_{t} | H | V_{p} | Z | S | S_{pr} | t_{pr} | B |
---|---|---|---|---|---|---|---|---|

Measurement units | kgf/mm^{2} | – | mm/min | μm | m/min | mm/min | mm | mm |

Minimum value | 20 | 1.38 | 0.10 | 16 | 20 | 60 | 0.01 | 20 |

Maximum value | 120 | 1.60 | 0.15 | 40 | 100 | 300 | 0.03 | 120 |

Parameter change increment | 10 | 0.02 | 0.01 | 8 | 5 | 10 | 0.01 | 2 |

σ_{t} | H | V_{p} | Z | S | S_{pr} | t_{pr} | B | P_{z} |
---|---|---|---|---|---|---|---|---|

kgf/mm^{2} | – | mm/min | μm | m/min | mm/min | mm | mm | H |

20 | 1.40 | 0.11 | 25 | 65 | 190 | 0.01 | 110 | 45 |

60 | 1.46 | 0.14 | 16 | 35 | 260 | 0.01 | 30 | 24 |

40 | 1.54 | 0.13 | 32 | 100 | 300 | 0.02 | 100 | 57 |

80 | 1.56 | 0.11 | 40 | 95 | 60 | 0.02 | 120 | 83 |

20 | 1.52 | 0.13 | 16 | 35 | 60 | 0.03 | 30 | 17 |

Unitless Parameters | ||||||||

${\widehat{x}}_{1}$ | ${\widehat{x}}_{1}$ | ${\widehat{x}}_{1}$ | ${\widehat{x}}_{1}$ | ${\widehat{x}}_{1}$ | ${\widehat{x}}_{1}$ | ${\widehat{x}}_{1}$ | ${\widehat{x}}_{1}$ | $\widehat{y}$ |

0.167 | 0.875 | 0.733 | 0.625 | 0.650 | 0.633 | 0.333 | 0.917 | 0.294 |

0.500 | 0.913 | 0.933 | 0.400 | 0.350 | 0.867 | 0.333 | 0.250 | 0.157 |

0.333 | 0.963 | 0.867 | 0.800 | 1.000 | 1.000 | 0.667 | 0.833 | 0.373 |

0.667 | 0.975 | 0.733 | 1.000 | 0.950 | 0.200 | 0.667 | 1.000 | 0.542 |

0.167 | 0.950 | 0.867 | 0.400 | 0.350 | 0.200 | 1.000 | 0.250 | 0.111 |

Parameter | α | β_{1} | β_{1} | β_{1} | β_{1} | β_{1} | β_{1} | β_{1} | β_{1} |
---|---|---|---|---|---|---|---|---|---|

Predicted value | 2.315 | 0.337 | 0.256 | 0.932 | −0.051 | −0.072 | −0.072 | −0.025 | 0.985 |

Actual value | 2.254 | 0.342 | 0.258 | 0.945 | −0.051 | −0.072 | −0.072 | −0.026 | 1.000 |

Relative error, % | 2.7 | 1.4 | 0.9 | 1.4 | 0.8 | 1.2 | 1.5 | 2.0 | 1.5 |

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**MDPI and ACS Style**

Pavlenko, I.; Saga, M.; Kuric, I.; Kotliar, A.; Basova, Y.; Trojanowska, J.; Ivanov, V.
Parameter Identification of Cutting Forces in Crankshaft Grinding Using Artificial Neural Networks. *Materials* **2020**, *13*, 5357.
https://doi.org/10.3390/ma13235357

**AMA Style**

Pavlenko I, Saga M, Kuric I, Kotliar A, Basova Y, Trojanowska J, Ivanov V.
Parameter Identification of Cutting Forces in Crankshaft Grinding Using Artificial Neural Networks. *Materials*. 2020; 13(23):5357.
https://doi.org/10.3390/ma13235357

**Chicago/Turabian Style**

Pavlenko, Ivan, Milan Saga, Ivan Kuric, Alexey Kotliar, Yevheniia Basova, Justyna Trojanowska, and Vitalii Ivanov.
2020. "Parameter Identification of Cutting Forces in Crankshaft Grinding Using Artificial Neural Networks" *Materials* 13, no. 23: 5357.
https://doi.org/10.3390/ma13235357