Accepted Manuscript Optimum surface roughness prediction for titanium alloy by adopting response surface methodology Aimin Yang, Yang Han, Yuhang Pan, Hongwei Xing, Jinze Li PII: DOI: Reference:
S2211-3797(16)30794-X http://dx.doi.org/10.1016/j.rinp.2017.02.027 RINP 590
To appear in:
Results in Physics
Received Date: Revised Date: Accepted Date:
31 December 2016 31 January 2017 17 February 2017
Please cite this article as: Yang, A., Han, Y., Pan, Y., Xing, H., Li, J., Optimum surface roughness prediction for titanium alloy by adopting response surface methodology, Results in Physics (2017), doi: http://dx.doi.org/10.1016/ j.rinp.2017.02.027
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Optimum surface roughness prediction for titanium alloy by adopting response surface methodology Aimin Yang1,*, Yang Han1, Yuhang Pan1, Hongwei Xing1, Jinze Li2 1. 2.
Hebei Key Laboratory of Data Science and Applications, North China University of Science and Technology, Tangshan, 063000,China College of Grassland and Environment Sciences, Xinjiang Agricultural University, Xinjiang 830052, China Email:
[email protected]
1
Optimum surface roughness prediction for titanium alloy by adopting response surface methodology
Abstract Titanium alloy has been widely applied in industrial engineering products due to its advantages of great corrosion resistance and high specific strength. This paper investigated the processing parameters for finish turning of titanium alloy TC11. Firstly, a three-factor central composite design of experiment, considering the cutting speed, feed rate and depth of cut, are conducted in titanium alloy TC11 and the corresponding surface roughness are obtained. Then a mathematic model is constructed by the response surface methodology to fit the relationship between the process parameters and the surface roughness. The prediction accuracy was verified by the one-way ANOVA. Finally, the contour line of the surface roughness under different combination of process parameters are obtained and used for the optimum surface roughness prediction. Verification experimental results demonstrated that material removal rate (MRR) at the obtained optimum can be significantly improved without sacrificing the surface roughness. Key words: titanium alloy; surface roughness; response surface methodology; design of experiment; iso-surface 1. Introduction Titanium alloy has been widely applied in aerospace, shipbuilding, chemical, metallurgical, medical and other industries because of the advantages of great corrosion resistance, high specific strength, and outstanding high temperature performance [1-3]. As the industry developing, the loads on the parts tend to complex and the working condition is abominable, such as high speed, high temperature, high pressure and heavy load [4-7]. Therefore, surface roughness plays a significant role in usability, reliability and service life [8-9]. Improving the surface quality has gained a lot of attention. Many studies report the methods of improving the surface quality [10-13]. Base on numerical researches and experience of processing, cutting speed, feed rate and depth of cut and other processing parameters have important effects on surface roughness [14-18]. Hence, improving the surface quality and processing efficiency through optimizing the cutting parameters has been widely used. 2
Recently, numerous methods for surface roughness prediction and optimal parameters include response surface methodology, genetic algorithm, neural network [15,18]. Response surface methodology (RSM) is a statistical method which is widely used in robustness design for optimal cutting parameters. Based on RSM, main factor and optimal control ranges can be found by optimizing the characteristics of products [17,19-21]. Hence, through a series of experiments, the characteristic function can be reconstructed according to the test results based on RSM [21-23]. RSM can be used to optimize parameters effectively by fitting response surface to simulate the limit surface in real condition [24-26]. This paper conducted the experiments of turning TC11 and studied the principle of the cutting parameters on surface roughness. The prediction model was presented to predict the surface roughness and cutting parameters. Then the prediction model was verified by analysis of variance and suitability test. Finally, the response surfaces and contour lines were developed for optimization of cutting parameters. 2. Design of experiment 2.1 Central composite design Generally, composite design is a test plan that adds some specific samples on the basis of regression design points. As a result, it can greatly reduce the number of experiments. At the same time, it can make the design satisfy the orthogonality by adjusting the asterisk arm based on the first design. The three factors center combined design is consist of factor point, center point and the axial point. It is clarified as circumscribed central composite design (CCC), inscribe central composite design (CCI), and central composite face-centered (CCF). CCF just needs 3 levels for every variable while CCC and CCI need 5 levels. Besides, in the model coefficient estimation, the CCF estimation precision is the highest and it can ensure enough coefficient estimation precision for peace party term. Therefore, CCF is an easy combinatorial design with high precious. In this work, CCF design, shown in Figure 1, is adopted.
3
Fig. 1. Central composite face-centered design
2.2 The regression model As described above, cutting speed, feed rate and depth of cut have significant effects on surface roughness. In this work, secondary mathematics regression method is adopted to establish the relationship between the surface roughness and processing parameters. The influences of various processing parameters on surface roughness are accurately investigated. k
k −1
Y ' = Ra − ε = b0 + ∑ bi xi + ∑ i
i =1
k
k
∑ b x x + ∑b x ij i
j =i +1
j
ii i
2
(1)
i =1
where Ra represents surface roughness, Y ' represents the predicted value of surface roughness, ε represents the test error, b represent predicted coefficients, x represent the processing parameters, k denotes the dimensions of the design variable space. The aim of the test is to acquire a set of optimum processing parameters to obtain the best surface quality. While, in reality it is very difficult to acquire optimum processing parameters. Hence, the response surface method is adopted to analyze the effects of the factors on the response values. Then the combination of the processing parameters that is able to achieve an optimal predicted value can be obtained. 2.3 The procedures of experiment In this work, titanium alloy TC11 is selected as the material. The length of the workpiece is L = 220 mm, and the diameter of the workpiece is D = 70 mm. The cutting Tool is carbide TiAlN coated. The rake angle and side angle are zero degree. All of the cutting tools used in this test are dry cutting tools. The Tokyo SU RFCOM1400D rough degree instrument was used to measure the values of surface roughness of the 4
workpieces. Assuming that the range of cutting speed vc ( x1 ) is between 60 and 140 m/min, the range of cutting depth a p ( x2 ) is between 0.25 and 0.75 mm, the range of feed f ( x3 ) is between 0.05 and 0.15 mm/r. z1 , z0 and z−1 represent the processing parameters of variable levels 1, 0 and -1, respectively. Then three variables are coded based on Eq. (2). xi =
zi − z0i , ∆i
(2)
i = 1, 2,3
where, xi represent the code of i th variable, zi represents the i th processing parameters, z0 i represents 0 level of processing parameters. ∆ i represent the interval range, which can be calculated as ∆ i = ( z1i − z0i ) / r ( r=1 in CCF). The Coding results of processing parameters levels are shown in Table 1. Based on Table 1, the turning test was conducted 20 times and the surface roughness was measured. The results are summarized in Table 2. Tab.1. Levels of the independent variables and coding identifications Factors
Code
-1
Levels 0
1
vc /( m/min)
x1
60
100
140
a p /mm
x2
0.25
0.50
0.75
f /( mm/r)
x3
0.05
0.10
0.15
Tab.2. The experimental results 5
No.
vc /( m/min)
a p /(mm)
f /(mm/r)
x1
x2
x3
Ra /(µm)
1
140
0.75
0.15
1
1
1
1.255
2
140
0.75
0.05
1
1
-1
0.146
3
140
0.25
0.15
1
-1
1
0.921
4
140
0.25
0.05
1
-1
-1
0.198
5
60
0.75
0.15
-1
1
1
0.963
6
60
0.75
0.05
-1
1
-1
0.309
7
60
0.25
0.15
-1
-1
1
0.985
8
60
0.25
0.05
-1
-1
-1
0.254
9
140
0.50
0.10
1
0
0
0.512
10
60
0.50
0.10
-1
0
0
0.593
11
100
0.75
0.10
0
1
0
0.546
12
100
0.25
0.10
0
-1
0
0.437
13
100
0.50
0.15
0
0
1
0.899
14
100
0.50
0.05
0
0
-1
0.180
15
100
0.50
0.10
0
0
0
0.450
16
100
0.50
0.10
0
0
0
0.449
17
100
0.50
0.10
0
0
0
0.451
18
100
0.50
0.10
0
0
0
0.450
19
100
0.50
0.10
0
0
0
0.450
20
100
0.50
0.10
0
0
0
0.449
3.
The analysis of the experimental results The experimental factors and surface roughness are converted into matrix form by
the test parameter transformation. The least squares method was used to obtained the coefficients of Eq. (1). Then the multivariate regression empirical formula between surface roughness and cutting parameters was established. The coefficients of Eq. (1) are as follows: Y ' = 0.458 − 0.007 x1 + 0.042 x2 + 0.393 x3 + 0.031x1 x2 + 0.056 x1 x3 + 0.039 x2 x3 + 2 1
2
0.083 x + 0.022 x2 + 0.070 x3
(3)
2
The analysis of variance for Eq. (3) is made, and the suitability of the regression model is tested. Table 3 shows the variance analysis of the regression model ( α is set as 0.05). S is the sum of squares (regression factor) between groups and sum of squares within group (residual error); M is the mean square, reflecting the ratio 6
between the sum of squares and degrees of freedom. The critical value can be obtained from the F distribution table, where F − tab = F0.05 (9,10) = 3.02 . Because F = 56.30 > 3.02 ,
it is concluded that the significant degree is obvious. Tab.3. ANOVA results for the regression model Degree of freedom
S
M
F
F-tab
9 10 19
1.726 0.034 1.76
0.192 0.003
56.3
3.02
Regression Residual error SUM
The accuracy of the regression model is on the base of the Rankit figure. If the Rankit figure is close to the linear relationship, the regression model is desirable, otherwise, it is inappropriate. Figure 2 shows Rankit plot of the model residual analysis. It can be seen that the correlation coefficient between residual sequence and expectations is R 2 = 0.952 , which is close to the linear relationship. As a result, the regression equation model is desirable. It also illustrates that the effectiveness of regression model and the credibility of analysis of variance.
Fig. 2. Rankit plot of regression model Before optimization of the processing parameters, the effect of the processing parameters should be analyzed. After dimensionless coding of the processing parameters, it shows that coefficient b and the coefficients of interaction and square in the regression model is irrelevant. However, the quadratic regression coefficients correlate with each other. Therefore, the influences of processing parameters on surface roughness can be determined based on the effect analysis of processing factors and marginal effect analysis. The two factors in the regression model are fixed at the zero level or other levels to obtain a single factor model. In this work, the sub models are obtained by fixing the 7
two factors of the regression equation on the zero level. The sub models of cutting speed, cutting depth and feed are as follows, y1 = 0.458 − 0.007 x1 + 0.083 x12 y2 = 0.458+ 0.042x2 + 0.022x2 2
(4)
y3 = 0.458+ 0.393x3 + 0.070x32
The marginal equations can be obtained after taking a derivative with respect to each single factor model respectively. The obtained marginal equations are as follows, y′1 = −0.007 + 0.165 x1 y′2 = 0.042 + 0.043x2
(5)
y′3 = 0.393 + 0.140x3
Figure 3 demonstrates the regression curve of the sub-model. It can be seen that x3 has significant positive effect on Ra , the value of Ra will increase greatly as the
value of x3 increases. However, Ra is not sensitive to x2 . The curve of x1 is a parabola, which is pointing up. When the value of x1 is equal to 0.044, the value of Ra reaches to the minimum value.
Fig. 3. The regression curve of sub-model Figure 4 shows marginal effects of the single factor. It indicates that x1 and x3 have significant impacts on the marginal roughness, while x2 has less effect. At the same time, all the three processing parameters have positive effects on the marginal roughness. That is to say increasing three processing parameters, the marginal roughness will increase. In addition, the factors at different levels have different 8
influence on the Ra . Hence, optimization is needed to determine the actual optimal processing parameters.
Fig. 4. The marginal effects of factors
4.
The optimization of the processing parameters The above analysis shows that reasonable processing parameters should be
chosen according to the effect of various factors on the Ra . Therefore, the distribution space of parameter can be obtained using response surface methodology, and then the optimal processing parameters can be chosen considering the factor effect and processing efficiency at the same time. Matlab is adopted to make the isosurfaces of different values of Ra in Eq. (3). The isosurfaces of different values of Ra are ploted in Figure 5. Figure 5 demonstrates that on the isosurface of Ra = 0.2 µ m , x3 falls into [−1, −0.75] , while on the isosurface of Ra = 0.4µ m , x3 falls into [0.7,1] . This shows
that when cutting the TC11, for a particular value of Ra , the range of x3 is the narrowest, x1 is the next. The desiable combination of x2 can be always chosen in the tests. It illustrates that x3 had the largest effect on Ra , followd by x1 . Therefore, to ensure the value of Ra , the range of the largest effect factor should be determined firstly. For factor x2 , it can be chosen considering the machining efficiency. The 9
material removal rate can be defined as, Q=
π Dnfa p
(6)
1000
where, n represents the speed of spindle. Then, it can be obtained from Eq. (3) and Table 1, Y ′ = 1.077 − 0.015vc − 0.797 a p − 2.048 f + 3.133 × 10 − 3vc a p + 0.028vc f + 3.086a p f + 2
2
5.172 × 10 − 5vc + 0.345a p + 27.900 f
(7)
2
Fig. 5 Iso-surfaces for different surface roughness values in 3-dimensional factorspace In the process of turning, the cutting efficiency can be improved through the optimization of cutting parameters without increasing surface roughness. From the above analysis, the scope of x 3 should be determined to ensure the value of the Ra . Then, the appropriate x 1 and x 2 should be selected to improve the value of Q . Assuming that x 3 was set at 0 level, namely, f = 0.1mm r , the contour map of Ra and Q can be obtained, which are shown in Figure 6.
10
Fig. 6. Contours of surface roughness and metal removal rate at 0.1mm/r feed rate
Figure 6 shows that the contours of Ra = 0.489µ m and Q = 3, 4.5 , and 6 cm3 / min intersect at A, B, C and D, indicating that these four groups of processing parameters have obtained the same Ra . While the machining efficiency at C and D are much higher than those of A and B. As a result, the processing parameters at C and D are better than those of A and B. In addition, reasonable parameters can be selected to get the best surface quality under the same processing efficiency. The contour line of Q = 4.5cm3 / min
shows that Ra of E and F are smaller than that of B, G, and H.
What’s more, from points C, D and E, it is concluded that the same Ra and Q can be obtained although the processing parameters are different. Therefore, the optimal processing parameters can be ultimately determined considering the specific production conditions in the actual processing process. Three groups of parameters were selected to conduct verification test through the analysis of Figure 5 and 6. The results are summarized in Table 4. Tab. 4. Results of verification tests No.
vc
(m/min) 1 2 3
110 110 80
ap
f
Ra
(mm) (mm/r) Experimental value Empirical value 0.25 0.25 0.5
0.05 0.1 0.1
0.135 0.426 0.479
0.132 0.433 0.482
As can be seen from Table 4, the empirical formula of the surface roughness 11
established using design of experiment and regression analysis has higher credibility. The changes of the first group and the second group show that the feed processing parameters has a positive impact on surface roughness. Therefore, determining the range of feed at first can ensure a desirable surface roughness. The test results of the second and the third groups of processing parameter show that the RSM is effective in the optimization of process of parameter. 5.
Conclusion (1) The model of the surface roughness was developed and the effects of various
factors were analyzed. It shows that the feed has the most important effect on surface roughness, followed by cutting speed. The cutting depth has minimal effect. (2) The distribution area of the processing parameters can be quickly determined based on the isosurfaces plot of surface roughness. Based on the analysis, the distribution area of the most important factors (feed) is the narrowest; the distribution area of the less important factors is much wider. It shows that the main factor is determined at first to ensure the surface roughness, and then adjust the smallest effect factors to guarantee the machining efficiency. (3) The contour map of small effect factor was plotted based on the surface roughness and material removing rate after the main factors were determined. It can be seen that the optimal processing parameters are often focused on the lower right of the contour map. At the same time, the optimal processing parameters are ultimately determined according to the specific production conditions. Reference 1.
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