# Surface Quality Evaluation in Orthogonal Turn-Milling Based on Box-Counting Method for Image Fractal Dimension Estimation

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## Abstract

Turn-milling is a relatively new cutting process combining the both advantages of turning and milling, during which the tool and the workpiece rotate simultaneously. Due to the small axial cutting force, turn-milling is especially suitable for the machining of miniature parts. Usually, the surface quality is characterized by roughness, but the size of miniature parts is difficult to meet the requirements of the sampling length, which will have a huge influence on the characterization results. This paper introduces the box-counting method for image fractal dimension estimation to evaluate the surface quality of miniature parts during orthogonal turn-milling. The fractal dimension is an independent parameter for scale, which is not influenced by the sampling length. The effect of cutting parameters on surface quality of miniature parts during turn-milling is also studied.

## Keywords

Surface quality Turn-milling Box-counting method Fractal dimension## 1 Introduction

Turn-milling is an emerging cutting technology, which has expanded the application ranges of mechanical processing. Depending on the position of the rotation axis between the tool and the workpiece, turn-milling can be divided into two different ways: orthogonal turn-milling and coaxial turn-milling. The relationship between the two rotation axes is vertical in orthogonal turn-milling while parallel in coaxial turn-milling. Orthogonal turn-milling is an effective method for external machining of the workpiece, especially for miniature parts, which is because of the small axial cutting force. Karagüzel et al. [1] have established the geometric, kinetic and mechanical models of orthogonal turn-milling and coaxial turn-milling. Egashira et al. [2] developed a turn-milling machine tool for small parts, which can machine microshafts with diameter of 50 μm. The surface morphology of parts machined by turn-milling is so complex that many researchers have studied the mechanism. Zhu et al. [3] have studied the surface topography of the workpiece during the process of orthogonal turn-milling. Vikram et al. [4] studied the influence of cutting parameters on the surface roughness of turn-milling. However, the method of using conventional roughness to characterize the surface quality of microparts is no longer appropriate, which is because the size of miniature parts is difficult to meet the requirements of the sampling length.

Fractal theory is a new developing subject of nonlinear problems, which is an effective tool for analysis of irregular things and phenomena with the self-similarity structure. Some fractal figures have exact self-similarity structure such as Koch curve, while some fractal figures have statistically self-similarity structure such as cutting surfaces. Fractal theory breaks the constraints of traditional Euclidean geometry, holding that the existence of things in nature is not limited to integer topological dimensions, but also has fractal dimension, which represents the complexity of fractal geometry. Jiang et al. [5] analyzed the features of the common machined surfaces with fractal geometry and proposed a two-dimensional method of Fourier transformation to calculate the fractal dimensions. El-Sonbaty et al. [6] used fractal theory to predict the surface topography of the workpiece during milling. Zhang et al. [7] studied on the application of fractal theory in grinding surface, who found that fractal dimension is more suitable to describe the surface topography than surface roughness, especially in miniature parts.

This paper introduces the box-counting method for image fractal dimension estimation to evaluate the surface quality of miniature parts during orthogonal turn-milling. The fractal dimension is an independent parameter for scale, which is not influenced by the sampling length. The effect of cutting parameters on surface quality of miniature parts during turn-milling is also studied.

## 2 Box-Counting Method for Image Fractal Dimension Estimation

The key problem of fractal theory is the calculation of fractal dimension. Many methods have been used such as power spectrum method, structure function method and box-counting method [8, 9, 10].

Box-counting dimension is one of the most widely used fractal dimension, which many scholars have studied and applied to a lot of fields. The mathematical basis of box-counting dimension is the Hausdorff dimension. And the calculation process is as follows:

*F*is overlapped by boxes of side

*r*, and the number of the boxes is

*N*

_{ r }(

*F*). If the set

*F*exhibits fractal characteristic, the relationship between

*N*

_{ r }(

*F*) and

*r*is expressed as:

*D*is the fractal dimension. And

*D*can be calculated by:

*M*×

*N*as a three-dimensional spatial surface with (

*x*,

*y*) representing the pixel position and

*z*representing the pixel gray-scale value. The image is overlapped by boxes of size

*r*×

*r*×

*h*, where

*M*/

*r*=

*G*/

*h*, and

*G*is the total number of gray levels, which is shown in Fig. 1.

*i*,

*j*) block fall into the box

*p*and

*q*, the number of boxes covering this block is counted as:

*n*

_{ r }(

*i,j*) is given by:

*I*

_{ k }(

*k*= 1, 2,…,

*n*) is the gray-scale value of the pixel in this block.

*n*

_{ r }(

*i,j*), where the boxes are in unfixed position, as shown in Fig. 2b. Thus, the calculation of the number of the boxes is the exact actual need, and the calculation is given by:

*N*

_{ r }(

*F*) can be obtained:

And different values of *N*_{ r }(*F*) can be obtained by changing the scale *r*. Take *logr* as the horizontal coordinate *X*, and take log*Nr(F)* as the longitudinal coordinate *Y*. Then a straight line is obtained by fitting the scatter points (log*r*, log*Nr(F)*). Finally the fractal dimension *D* is obtained by calculating the slope of the straight line.

## 3 Experiments Details

Main physical properties of 7075 aluminum alloy

Elastic modulus | Poisson ratio | Density | Yield strength | Brinell hardness | Elongation rate |
---|---|---|---|---|---|

71 (Gpa) | 0.33 | 2.81 (g/cm | 500 (Mpa) | 150 | 11 |

Parameters of the experiments

Number | Rotation speed of cutting tool (r/min) | Rotation speed of workpiece (r/min) | Depth of cut (mm) | Axial feed rate (mm/r) | Tool diameter (mm) |
---|---|---|---|---|---|

1 | 1000, 2000, 3000, 4000 | 10 | 0.2 | 0.2 | 1 |

2 | 2000 | 5, 10, 15, 20 | 0.2 | 0.2 | 1 |

3 | 2000 | 10 | 0.1, 0.15, 0.2, 0.25 | 0.2 | 1 |

4 | 2000 | 10 | 0.2 | 0.1, 0.15, 0.2, 0.25 | 1 |

## 4 Results and Discussion

*S*

_{1}= 2000 r/min,

*S*

_{2}= 10 r/min,

*a*

_{p}= 0.2 mm,

*f*= 0.2 mm/r) is chosen as an example to calculate the fractal dimension of the machined surface. The surface topography gray image of the microshaft is shown in Fig. 5, from which the surface topography is seen clearly.

*r*is a key problem to deal with, which cannot be too large or too small. If

*r*is too large, the number of the boxes cannot exhibit the complexity of surface morphology; and if

*r*is too small, the number of the boxes does not exhibit fractal characteristic. The appropriate scale of the box size

*r*is from M/8 to M/25, which is based on experimental experience. When the box size

*r*is chosen M/10, M/15 and M/20, the numbers of the boxes that overlap all pieces are shown in Fig. 6, which reflects the complexity of the surface topography.

As shown in Fig. 7, the correlation coefficient *R* is − 0.9993, which means lg*r* and lg*N*_{ r }(*F*) show a high degree of linear correlation. And the fractal dimension *D* is 2.6332, between 2 and 3, which has verified the surface topography of turn-milling exhibits fractal characteristic.

As shown in Fig. 8, when the rotation speed of cutting tool increases from 1000 to 3000 r/min, the fractal dimension of the machined surface decreases from 2.72 to 2.52, when the rotation speed of workpiece increases from 5 to 15 r/min, the fractal dimension increases from 2.54 to 2.77, when the depth of cut increases from 0.1 to 0.3 mm, the fractal dimension increases from 2.61 to 2.64, when the axial feed rate increases from 0.15 to 0.25 mm/r, the fractal dimension increases from 2.41 to 2.73, which shows the rotation speed of cutting tool, the rotation speed of workpiece and the axial feed rate have large influence on the surface topography, and the depth of cut has a little influence on the surface topography.

## 5 Conclusion

- (1)
By using box-counting method for image fractal dimension estimation, the surface quality of miniature parts is better evaluated than the conventional roughness method. And the surface fractal dimension value of orthogonal turn-milling is from 2.41 to 2.77.

- (2)
The effect of turn-milling parameters on surface quality of miniature parts is obtained through experiments, which shows the fractal dimension is influenced mainly by the rotation speed of workpiece, the rotation speed of cutting tool and the axial feed rate, rarely by the depth of cut.

- (3)
To obtain better surface quality with small fractal dimension, the rotation speed of workpiece and the axial feed rate should be lower, the rotation speed of cutting tool should be higher, and the depth of cut should be chosen properly.

## Notes

### Acknowledgements

This work has been supported by Natural Science Foundation of China (No. 51575050 and No. 51505034).

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