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Advanced Materials Research Vols. 154-155 (2011) pp 856-861 Online available since 2010/Oct/27 at www.scientific.net ? (2011) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/AMR.154-155.856

Investigation of Surface Roughness Prediction in Single-Point Diamond Turning Qingliang Zhaoa, Junyun Chenb, Jian Luoc

Center for Precision Engineering (CPE), Harbin Institute of Technology, Harbin, China azhaoqlyx@126.com, bsophiacjy@gmail.com, crhythmlj@163.com Keywords: prediction model, surface roughness, vibration, single-point diamond turning

Abstract. Experimental results indicate the previous theoretical model cannot predict well the surface roughness in single-point diamond turning on a precision lathe. In solving that, an improved model was presented in this paper. The difference between the previous model and the improved model is that the relative tool-workpiece vibration is measured before cutting operation using a capacitive displacement sensor in the previous model whilst the vibration is extracted from the measured surface profile in the improved model. The relative vibration was first studied under various cutting conditions to establish the vibration modes under corresponding cutting conditions. Then the surface roughness was predicted based on the vibration modes. The results prove that there is good agreement between the predicted values and measured values and the improved model is useful and reliable. Introduction To realize a deterministic surface roughness value in a machined part is increasingly required in manufacture of high precision components especially for mould inserts of injection moulding lenses. Among various machining process, Single-point diamond turning (SPDT) is a preferable process to produce components with surface roughness in nanometer range [1-2]. Therefore, it is significant to investigate the prediction model of surface roughness in order to control the surface roughness in a desired range in SPDT. Many research works have been reported in surface roughness generation in order to predict the machined surface quality. Takasu et al. [3] established a two-dimensional theoretical model of surface roughness through introducing the relative vibration between the tool and the workpiece into the ideal roughness profile. He found that surface roughness in tool feed direction is more dominant than that in the main cutting direction. On the basis of the two-dimensional theoretical model, Lee et al. [4-5] established a three-dimensional theoretical model of surface topography. In this model, the machining parameter of the tool geometry and the relative motion between the tool and the workpiece were used to characterize the kinematics of diamond turning process. The topography was generated by a linear mapping of the predicted surface roughness profiles on the surface elements of a cross lattice. Besides the investigation of the theoretical model, Pandit et al. [6] analyzed the surface roughness formation using the data dependent system approach. Sata et al. [7] studied the process of surface roughness generation in turning operation using spectral analysis approach. However, previous research assumed that the cutting process is orthogonal and the workpiece materials are homogeneous and isotropic. Moreover, the relative vibration between the tool and the workpiece is only considered in the tool feed direction as a steady simple harmonic motion with a low frequency and a small amplitude. In this study, a series of cutting experiments was conducted to study the prediction of surface roughness in SPDT based on the theoretical model of surface roughness generation. But the results show that the model is not capable of predicting well the surface roughness. In solving that, an improved model is presented to predict and control the surface roughness in SPDT with the aid of spectral analysis approach.

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Experiments A series of face cutting tests of SPDT were conducted on a two-axis CNC precision lathe (homemade at Harbin Institute of Technology, China). The specimen material used in the tests is electroless nickel phosphorus, an excellent mould insert material, which was electroless nickel plated on a plate aluminum alloy (7075). The specimen had a plating depth of more than 50?m and a hardness of more than 50HRC. The cutting conditions were as follows: (1) tool nose radius was set as 0.50mm; (2) spindle speeds were set as 100r/min, 200r/min,300 r/min and 400r/min; (3) feed rates were set as 20?m/s, 30?m/s, 40?m/s and 50?m/s; (4) depths of cut were set as 4?m, 6?m 8?m and 10?m. A Taylor-Hobson profilometer (aspherics measurement system PGI1240) was used to measure the surface roughness profile of the machined surface. Theoretical model and spectral analysis Theoretical model. In the model, the relative vibration between the tool and the workpiece is assumed as invariable in cutting operation and it is usually measured by a capacitive displacement sensor when running the lathe but not cutting workpiece. The frequency f z of relative vibration and the spindle speed n is defined as:

60 f z = ?+ε n

(1)

Where ? is 0 or a positive integer and ε is a decimal fraction in the range ? 0.5 ≤ ε ≤ 0.5 . Then a phase shift is defined as: (2) φ = 2πε In X-Z plane (the X-axis along the tool feed, the Z-axis along the infeed cutting direction), the tool locus can be expressed as:

x Z t ( x) = A[1 ? cos(φ )] s

(3)

where zt(x) is the relative displacement between the tool and the workpiece and A is the amplitude of the vibration. If the tool tip is taken at the origin of X-Z plane, the cutting edge profile can be expressed as:

Z ( x) = R ? R 2 ? x 2 ≈ x2 2R

(4)

Assume the first tool profile starts at the lowest point being taken at the perimeter of a workpiece with diameter D. the number of successive movement of the tool along the feed in one machining cycle is equal to D/2s. The tool locus can be rewritten as: (5) Z t (i ) = A{1 ? cos[(i ? 1)φ ]} for i = 1,2,..., N The cutting edge of the ith tool profile and the i+1th tool profile can be derived as:

Z i ( xi ) = A{1 ? cos[(i ? 1)φ ]} + [ xi ? (i ? 1)s]2 2R ( x ? is ) 2 Z i +1 ( xi +1 ) = A[1 ? cos(iφ )] + i +1 2R

(6) (7)

with i = 1,2,..., N From Eqs. (6) and (7), the intersecting point of the ith tool profile and the i+1th tool profile ( xi ,i +1 , H i ,i +1 ) can be expressed as:

4 RA sin[(i ? 1 / 2)φ ] sin(φ / 2) + (2i ? 1) s 2 for i = 1,2,..., N 2s {4 RA sin[(i ? 1 / 2)φ ] sin(φ / 2) ? s 2 }2 for i = 1,2,..., N H i ,i +1 = A[1 ? cos(iφ )] + 8Rs 2 xi ,i+1 =

(8) (9)

The surface roughness profile was constructed by trimming the lines above the points of intersection [3-4].

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Spectral analysis. Spectral analysis can be used to extract the feature of surface roughness profile to gain a better understanding. By taking N as the number of sampling points in measured length l, l s is equal to l/N and z (nl s ) represents the surface height along the infeed cutting direction. The amplitude spectrum of the roughness wave along the tool feed direction can be defined as:

N ?1

P (v k ) = ∑ z (nl s ) exp(?2πjnl s v k )

n =0

for n = 0,1,2,..., N ? 1

(10)

Where vk is the frequency of the roughness wave which expresses the number of the wavelength

λ k within a unit length. The power spectral of the roughness wave is achieved by discrete Fourier

transformation computed with a fast Fourier transformation algorithm [7]. Experimental results analysis based on above methods. The relative tool-workpiece vibration on the used lathe in experiments was first measured by a capacitive displacement sensor. A mode of vibration with amplitude of 1.06?m and a frequency of 13.0Hz was measured. Then, the surface roughness profile was predicted under a cutting condition according to above methods. When the cutting condition was set as a spindle speed of 300r/min, a feed rate of 50?m/s and a depth of cut of 10?m, the surface roughness profile was predicted by considering ideal surface profile, the relative vibration between the tool and the workpiece as well as the tool interference. Fig.1 (a) shows the predicted results. The spectral analysis result indicates that the mode of vibration has 1.00?m amplitude and 40Hz frequency. But the roughness profile of measured surface has obvious difference when compared to the predicted one, as shown in Fig.1 (b). From the corresponding spectral analysis result, there are two vibration amplitudes of 0.132?m and 0.095?m at the frequencies of 5Hz and 27.5 Hz respectively on the measured surface. The results indicate that the previous methods can not predict well the surface roughness of the machined specimens in this paper’s experiments. So it is necessary to present an improved method to predict the surface roughness. 原原原原

Surface height (?m) 表表表表（ um）

3 2 1 0

Power of amplitude (?m2) Surface height (?m) 能能 表表表表（ um）

1 0.5 0 -0.5 -1 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Power of amplitude (?m2) 能能

检检检 度 (mm) Measured length (mm) 频频频 (频 域 )

2 1.5 1 0.5 0 0 50 100 150 200 250 300 350 400 450

检检检 度 (mm) Measured length (mm) 频频频 (频 域 )

0.02

0.015 0.01

0.005 0

0

10

20

30

40

50

60

70

80

90

100

Frequency (Hz) (a) (b) Fig. 1 (a) The predicted surface roughness profile and power of amplitude; (b) The measured surface roughness profile and power of amplitude.

频 率 (Hz) Frequency (Hz)

频 率 (Hz)

Improved model for predicting the surface roughness Improved model. Using the previous methods the surface roughness cannot be predicted well. It is mainly due to the following factors: 1. The vibration mode in the previous method was measured under the condition before cutting operation; 2. It is possible that there are great difference between the measured vibration and the real relative vibration in cutting operation; 3. The real relative vibration changes with the cutting condition changes. In solving that, the relative vibration used for predicting the surface roughness is also considered as a simple harmonic motion, but it is gained from the real cutting operation in the improved model. However, it is well known that to measure the relative vibration between the tool and the workpiece is

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very difficult in real cutting operation. In this model the spectral analysis method was used to extract the real relative vibration from the measured surface profile, as shown in Fig. 2. The key of the improved model is to know the relative vibration under different cutting conditions on the used lathe. A vibration mode of simple harmonic motion includes vibration amplitude and frequency. But the surface roughness is mostly decided by the vibration amplitude. If a database of the relative tool-workpiece vibration under different cutting conditions is first established through a lot of cutting tests on a lathe, the improved model will become significative and reliable. Therefore, the relative vibration under different cutting conditions should be first studied.

machined surface to measure the roughness profile to process the data tool interference real relative vibration ideal surface profile predicted surface profile and surface roughness

Fig. 2 The improved model for predicting the surface roughness The relative vibration under different cutting conditions. When the spindle speed was set as 300r/min and the depth of cut was set as 10?m, the effect of feed rate on the vibration is shown in Fig.3 (a). The vibration amplitude increases from 126nm to 132nm with the feed rate increases. The frequency remains 5Hz under different feed rates. Fig.3 (b) shows the effect of depth of cut on the vibration, when the spindle speed was set as 300r/min and the feed rate was set as 40?m/s. The vibration amplitude decreases a little with increasing depth of cut. The frequency is found to remain 5Hz with the change in depth of cut. It indicates that the relative vibration in cutting operation was almost unaffected by the change in feet rate and by the change in depth of cut. When the feed rate was set as 50r/min and the depth of cut was set as 10?m, the effect of spindle speed on the vibration is shown in Fig.3 (c). There is significant change in the vibration amplitude under different spindle speeds. The vibration amplitude decreases with increasing spindle speed at a low spindle speed and increases with increasing spindle speed at a high spindle speed. The frequency is found to be 5Hz at a small vibration amplitude and 10Hz at large vibration amplitude. It can be concluded that the vibration amplitude was remarkably affected by the change in spindle speed.

150

200

主频主主 （ (nm) Amplitude um）

140 130 120 110 100 30

主频主主（ (nm) Amplitudenm ）

180 160 140 120 100 4 5 6 7 8 9 10

35

40

进 进 速 速 (um/s)

45

50

Feed rate (?m/s)

1000

(a)

Amplitude (nm)

800 600 400 200

背背背背 (um) Depth of cut (?m)

(b)

0 100

150

200

250

300

350

400

(c) Fig.3 (a) effect of feed rate on the vibration; (b) effect of depth of cut on the vibration; (c) effect of spindle speed on the vibration. Among spindle speed, feed rate and depth of cut, it is found that spindle speed had the most influence on the vibration. Therefore, the effect of spindle speed on the vibration is only considered to

主 Spindle轴轴 主 (r/min) speed

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establish a series of vibration modes under different spindle speeds which was established according to the experimental results for the improved model. When predicting the surface roughness under a cutting condition, it only needs to find the vibration mode under corresponding spindle speed. Experimental verification for the improved model Using the improved model presented in this paper, the arithmetic roughness was predicted under different depth of cuts, when the spindle speed was set as 300r/min. As shown in Fig.4 (a), the predicted arithmetic roughness has a little increase with increasing depth of cut. That can be explained that the relative vibration between the tool and the workpiece becomes the dominant component to construct the surface roughness profile under the cutting condition. The measured values are also not found to change a lot with increasing depth of cut. It supports that the effect of depth of cut on the vibration can be neglected in the improved model. The difference between the predicted values and the measured values are found to be within 20 per cent in most cases under different depth of cuts. Fig.4 (b) shows the predicted arithmetic roughness and the measured arithmetic roughness with different feed rates. It can be seen that the predicted values have almost no change with increasing feed rates. However, the previous result shows that surface roughness increases with the feed rate increases. The difference between the predicted result in this paper and the previous result can be explained that the relative vibration is a main factor in the surface roughness profile under the cutting condition. The measured values have a little change with increasing feed rate. It also supports that the effect of feed rate on the vibration can be neglected in the improved model. The errors for the prediction of surface roughness are found to be within 10 per cent. The results show the difference between the predicted values and the measured values are within 20 per cent under various cutting conditions. Therefore, there is good agreement between the predicted result with the improved model and the measured result. The results also indicate that the improved model presented in this paper is proved to be useful and reliable for predicting surface roughness in SPDT.

150

Arithmetic roughness Ra (nm) 表表表表表（ nm ）

200

Arithmetic roughness Ra (nm)

145 140 135 130 125 120 115 110 105 100 30 35 40 45 50

Measured value 实实实

150

Model predicted value

预预预 实实预

预预实 Model predicted value

100

Measured value

50

4

5

6

7

8

9

10

(a) (b) Fig.4 (a) Effect of depth of cut on arithmetic roughness; (b) Effect of feet rate on arithmetic roughness. Conclusions An improved model for predicting surface roughness in SPDT was presented in this paper by considering ideal surface profile, tool interference and the real relative tool-workpiece vibration which was extracted from the machined surface using spectral analysis approach. Experimental results prove that the predicted surface roughness using the model has good agreement with the measured value and the model is useful and reliable for predicting surface roughness under various cutting conditions. Although the relative tool-workpiece vibration is measured under the condition before cutting in the previous theoretical model, the previous model can also predict well the surface roughness when the cutting tests are conducted on an ultra-precision diamond lathe. That is because the relative

背背背背 (?m) Depth of cut(um)

进进 速速 (um/s) Feed rate (?m /s)

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tool-workpiece vibration on an ultra-precision lathe is approximatively invariable under various cutting conditions. However, if the cutting experiments are conducted on a precision lathe, the previous model will be fail to predict the surface roughness due to variational relative tool-workpiece vibration when changing cutting condition on a precision lathe. The improved model presented in this paper can solve this problem well. Acknowledgements The authors would like to express their sincerely thanks to the CAST Innovation Fund (Contract No. CAST20081003) and the National Key Project (Contract No. 09ZX04001-151) for their financial support of the research work. References [1] D. Karayel: Journal of Materials Processing Technology, Vol.209 (2009), p.3125–3137 [2] C. F. Cheung and W.B. Lee: Journal of Machine Tools and Manufacture, Vol.41 (2001), p.851-875 [3] S. Takasu, et al.: Annals of the CIRP, Vol.34 (1985), p.463–467 [4] C. F. Cheung, W.B. Lee: Journal of Machine Tools and Manufacture, Vol.40 (2000), p.979-1002 [5] W.B. Lee, et al.: China Mechanical Engineering, Vol.11 (2000), p.845-848 [6] S.M. Pandit: Annals of the CIRP, Vol.30 (1981), p.487-492 [7] T. Sata, et al.: Annals of the CIRP, Vol.34 (1985), p.473-476

Materials Processing Technologies doi:10.4028/www.scientific.net/AMR.154-155 Investigation of Surface Roughness Prediction in Single-Point Diamond Turning doi:10.4028/www.scientific.net/AMR.154-155.856

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