signal analysis of surface roughness in diamond turning of lens molds

� Corresponding author. T

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E-mail address: hocheng@

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r# 2004 Elsevier Ltd. All rights reserved.

4.06.003

International Journal of Machine Tools & Manufacture 44 (2004) 1607–1618

www.elsevier.com/locate/ijmactool

Signal analysis of surface roughness in diamond turningof lens molds

H. Hocheng a,�, M.L. Hsieh b

a Department of Power Mechanical Engineering, National Tsing Hua University, Hsinchu 300, Taiwan, ROCb Mechanical Industry Research Laboratories, Industrial Technology Research Institute, Hsinchu, Taiwan, ROC

Received 10 February 2004; accepted 2 June 2004

Abstract

Diamond turning of high-precision lens molds is an important production process. The surface roughness of the mold heavilyaffects the quality of lens. In diamond turning, the surface roughness obtained depends on the cutting tool, the cutting conditions,the machine characteristics, the surrounding vibrations and the work piece material. This work studies the surface roughnessobtained from the diamond turning of a phosphor–bronze lens mold with various tool nose radii, spindle speeds, feed rates andcutting depths. The surface roughness was measured in the time domain using a Form Talysurf instrument (a stylus-type surfaceroughness meter) and then transformed into the frequency domain using the fast Fourier transform. Based on the magnitude ofthe intensity, the tool geometry, low-frequency vibration and the measuring instrument are identified as the main influencing fac-tors of the generated surface roughness. The intensities associated with the latter two vary little with the cutting conditions andare thus considered constant. The intensity of the tool geometry varies with the feed rate, the spindle speed and the radius of thetool nose. A relationship between the root-mean-square summation of the surface roughness and cutting conditions was found.The model agrees well with the experimental results. The analysis also identified the critical feed rate that maximized machiningproductivity, below which the surface roughness was only slightly improved as the production rate fell sharply.# 2004 Elsevier Ltd. All rights reserved.

Keywords: Diamond turning; Surface roughness; Feed rate; Nose radius; Fast Fourier transformation; Frequency spectrum

1. Introduction

In recent years, optical lenses have become key

components in volume consumer products, such as

digital cameras. Surface roughness is one of the most

important factors in evaluating the quality of a lens.

Continuous improvement in machining precision has

enabled the application of ultra-precision cutting [1],

achieving high accuracy and good surface roughness.

Ultra-precision diamond turning is extensively used in

manufacturing high-precision optical lenses with a sur-

face roughness of a few nanometers [2,3]. The surface

roughness obtained by diamond turning depends on

the cutting tool, the cutting conditions, the machine

characteristics, the surrounding vibrations and thework piece material [4–6].Diamond turning, a micro-machining process, cuts

to a depth of several micrometers only. The minimumdepth of the cut is a function of the tool edge radius[7]. A new diamond tool has an edge radius of about0.2 lm, and cuts to a minimum depth of about 40 nm.For a shallow cut and a low feed rate, the cutting forceis commonly several newtons [8–10]. The size effect onthe cutting force is larger than in traditional machining[11,12]. A blunt tool edge produces worse surfaceroughness than a sharp tool edge [7,13]. The tool-workvibration also increases the surface roughness [6,14].The cutting temperature during diamond turning isabout 170

vC [15], and the life of the diamond tool is

measured by its cutting length of over 1000 km [16].Many theoretical and practical studies [17–22]

have conducted investigated surface roughness. Many

1608 H. Hocheng, M.L. Hsieh / International Journal of Machine Tools & Manufacture 44 (2004) 1607–1618

methods have also been developed for measuring sur-face roughness. Among which include optical method[23,24] and stylus method [25,26]. The stylus methodhas widely been used to measure surface roughness, forgood resolution, ease of data output and operation.Pandit [18] used the data-dependent systems (DDS)

method to analyze the generation of surface roughnessduring turning. Sata et al. [27] applied a spectrumanalysis method to determine the roughness profile of aturned surface. Cheung used the power spectrummethod to evaluate the surface roughness analysis onan aluminum alloy [28]. The surface finish profile of a

workpiece has numerous periodic components—the

cutting feed component, the spindle rotational error

component, and the chatter vibration component.This study applied the stylus method to determine

surface roughness data and spectrum analysis to ident-

ify the factors that dominate surface roughness in dia-

mond turning. A surface roughness model is expected

to set the proper cutting conditions. Experiments on

face cutting were undertaken. An ultraprecision lathe

with a single crystal diamond tool was used to turn

phosphor–bronze in the experiment.

urface. (a)Measured surface profile, (b)measured and low-pass filtered surface profile, and (c) micro
Fig. 1. Machined s scopic photograph.

H. Hocheng, M.L. Hsieh / International Journal of Machine Tools & Manufacture 44 (2004) 1607–1618 1609

2. Components of surface roughness spectrum

In a turning operation, the revolution rate of thespindle, the feed rate and the radius of the tool nose allaffect the ideal surface roughness of the machined sur-face. The ideal average surface roughness (Ra1) is afunction of only the tool geometry that is regeneratedon the workpiece surface. The average ideal surfaceroughness can be approximated by [11]

Ra1 ¼ 0:0321� f 2=R ð1Þ

where f is the feed per workpiece revolution and R isthe tool nose radius.However, the real surface roughness is often found

to be worse than the above ideal surface roughness.For example, at a feed rate of 1 mm/min, a spindlerevolution rate of 1340 rpm and a tool nose radius of1.5 mm, ultraprecision machining yields an averageroughness of 0.0103 lm, whereas Eq. (1) predicts anideal surface roughness Ra1 ¼ 0:000012 lm. The differ-ence is extremely large. The average surface roughnessof 0.01 lm or less is required in mirror turning, so theabove simple model of the ideal surface roughness isnot sufficiently precise.In the turning process, the cutting tool edge and

vibrations induced by the machine tool and environ-mental vibration can affect the surface of the workpiece.Periodical marks of corresponding frequency are made;they combine eventually to form a complex wave on thesurface. Fig. 1 is a machined surface profile that wasmeasured by a Form Talysurf surface instrument. It canbe transformed into the frequency domain using the fastFourier transform. Fig. 2 plots the results of the trans-formation into the frequency domain.Fig. 1 presents the periodic profile that remains on

the machined surface. The waves are consideredto have been generated by the regular feeding of theturning tool. The effect is attributed to the geometry ofthe tool.The measuring instrument is an external high-fre-

quency factor that influences, as shown in Fig. 2. A

digital low-pass filter can be used to filter out the highfrequency, so the results can be compared with themicroscopic photograph.Fig. 1 presents a surface profile measured using the

Form Talysurf without and with a digital low-pass fil-ter. The high-frequency effects on the surface roughnessare clearly less significant than the main oscillation.The third factor that affects the surface roughness islow-frequency vibration from the surroundings. Themeasured profile agrees well with the microscopicphotograph of the workpiece.Fig. 2 indicates that the tool geometry, the low-

frequency vibration and the measuring instrument arethe three factors that dominate the surface roughnessduring turning. These three factors are independent,and so satisfy the root-mean-square relationship

IT ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiI21 þ I22 þ I23

qð2Þ

where IT is the sum of the intensities, and I1, I2, and I3are the intensities of the tool geometry, the low-frequency vibration and the measuring instrument,respectively.The low-frequency vibration does not vary signifi-

cantly with the feed rate or the tool nose radius and sois considered to be a constant. The intensity associatedwith the instrument is also considered to be constant.The effects of tool geometry on the surface roughnessvary with feed rate, spindle revolution and tool noseradius. Fig. 3 plots the relationship between the inten-sity and the feed rate.The sum of the root-mean-square surface roughness

can be expressed as the sum of the intensities, each ofwhich influences the RMS surface roughness.

RqT ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiRq21 þ Rq22 þ Rq23

qð3Þ

where RqT is the sum of the RMS surface roughness,and Rq1, Rq2 and Rq3 are the RMS surface roughnessof the tool geometry, low-frequency vibration and themeasuring instrument, respectively.

Fig. 2. FFT of machined surface profile.


An RMS surface roughness model was establishedfrom the components of the surface roughness spec-trum (Fig. 4). It provides the same results in Fig. 3.

3. Experiment

The experimental setup includes an ultraprecisionlathe (Fig. 5(a)). The single crystal diamond tool turnsthe lens mold of the phosphor–bronze by face cutting(Fig. 5(b)). Various tool nose radii, spindle revolutionrates, feed rates and cutting depths were used (Table 1).The surface roughness of the machined surfaces wasmeasured using a Form Talysurf Instrument. Table 2presents its specifications. The measurement can beused to calculate the arithmetic average of the surfaceroughness (Ra) and the RMS value (Rq). The FFTfunction in Matlab software transforms the surfaceprofile data into the frequency domain, along with theintensities of the influence factors.

4. Results and discussion

4.1. Effects of machining parameters

The machining parameters include the radius of thetool nose, the rate of revolution of the spindle, the feed

rate and the cutting depth, as shown in Table 1. Theexperiment yielded 270 surface roughness (Ra) data.Table 3 shows the ANOVA of the surface roughness,indicating that the spindle revolution, the tool noseradius and the feed rate are major influence parameters

Fig. 3. Relationship between intensity and feed rate.

Fig. 4. Relationship between RMS surface roughness and feed rate.

. Experimental setup. (a) Moore M18-AG ultraprecision l
Fig. 5 athe,
and (b) schematic diagram.

Table 1

Experimental condition

Machine tool M
oore M18-AG
Workpiece P
hosphor–bronze (C5191)
Cutting tool (mm) S
ingle crystal diamond, nose
radius: 1:02=0:52=0:23

Spindle revolution (rpm) 1
500=1000=800
Feed rate (mm/min) 1
=3=4=5=8=10=12=15=20=25=30
Cutting depth (mm) 0
:01=0:005=0:001
Table 2

RTH Form Talysurf specification

Straightness V
ertical direction: 0:25 lm=60 mm Measuring speed 1 .0 and 0.5 mm/s
Resolution 1
0 nm at 6 mm range
Stylus tip radius 2
lm Traverse length 1 20 mm


(significance ¼ 0), whereas the cutting depth is con-sidered insignificant. Fig. 6 plots the relationship

between the surface roughness and the machining para-

meters. The surface roughness increases by the square

of the feed rate, but decreases linearly as the rate of

revolution of the spindle and the radius of the tool

nose increase. Fig. 6(d) indicates that cutting depth

slightly affects the surface roughness. The results

of diamond turning machining are similar to those

obtained using traditional turning tools, as supported

by the spectrum analysis in the next section.

een surface roughness and machining parameters. (a) Relationship between surface
Fig. 6. Relationship betw roughness and feed rate,
(b) relationship between surface roughness and spindle revolution, (c) relationship between surface roughness and tool nose radius, and

(d) relationship between surface roughness and cutting depth.

Table 3

ANOVA of surface roughness

Source T
ype III sum
of squares

df M
ean square F S ig.
Model 0
.133 16 8 :283� 10�3 5 3.573 0 .000
A 6
:305� 10�3 2 3 :153� 10�3 2 0.391 0 .000
B 1
:045� 10�2 2 5 :226� 10�3 3 3.800 0 .000
C 1
:767� 10�5 2 8 :837� 10�6 0.057 0 .944
D 4
:522� 10�2 9 5 :024� 10�3 3 2.498 0 .000
Error 3
:927� 10�2 254 1 :546� 10�4 Total 0 .172 270
A: spindle revolution; B: tool nose radius; C: cutting depth;D: feed rate.

4.2. Spectrum analysis

Fig. 7 shows the frequency spectrum diagram of 10

workpieces machined at the same rate of revolution of

the spindle (1500 rpm), tool nose radius (0.52 mm) and

cutting depth (0.01 mm) but at different feed rates. The

diagram clearly shows the three dominant factors

identified in Section 4.1. These workpieces were cut at

the same rate of revolution of the spindle, so the effects

of the tool geometry were all at the same frequency

(25 Hz). The measuring speed of the instrument is con-

stant, so the frequency of the effects of the measuring

instrument increase with the feed rate as in Fig. 7. The

effects of low-frequency vibration remain constant and

weak.

Table 4 shows the ANOVA of the tool geometry

effects (I1). It reconfirms that the feed rate, the tool

nose radius and the rate of revolution of the spindle

are dominant factors. Tool geometry effects (I1) were

related only to tool geometry, so it can be considered

the static ideal minimum surface roughness due to

turning. The ideal surface roughness (Ra1) is given by

Eq. (1), in which it is affected only by the feed rate, the

tool nose radius and the rate of revolution of the spin-

dle, as shown by Table 4. Fig. 8(a) indicates that the

intensity I1 increases by the square the feed rate, and

the result is consistent with Eq. (1).Table 5 shows the ANOVA of the low-frequency

vibration effects (I2). The feed rate and tool nose radius

are significant. The low-frequency vibration was gener-

ated by the environmental vibration and the tool-work

vibration. The tool-work vibration is a high-frequency

vibration. In turning, this vibration causes a phase

error near its pitch. In the experiment, the surface

roughness was measured in the radial direction, so the

tool-work vibration became a low-frequency phenom-

enon. The tool-work vibration was excited by the cut-


instrument effects (I3) and the low-frequency vibrationeffects (I2) are kept nearly constant at a low level.

4.3. Verification of model

The fast Fourier transformation (FFT) transformsthe time-domain signals (obtained using the Talysurfsurface roughness instrument) into the frequencydomain to identify the major influencing factors. Eachinfluencing factor has its intensity in frequency domainand its surface roughness in the time domain.

ip between intensity and feed rate. (a) Tool geometry, (b) low-frequency vibrations, and (c) mea
Fig. 8. Relationsh suring instrument.
Table 5

The ANOVA of low-frequency vibration intensity

Source T
ype III sum
of squares

df M
ean square F S ig.
Model 4
:248� 10�3 16 2 :655� 10�4 265.579 0 .000
D 5
:999� 10�5 9 6 :666� 10�6 6.667 0 .000
C 2
:737� 10�6 2 1 :368� 10�6 1.369 0 .256
B 2
:980� 10�5 2 1 :490� 10�5 14.904 0 .000
A 5
:076� 10�7 2 2 :538� 10�7 0.254 0 .776
Error 2
:540� 10�4 254 9 :998� 10�7 Total 4 :502� 10�3 270
A: spindle revolution; B: tool nose radius; C: cutting depth; D: feed

rate.


Fig. 10(a) plots the regression of intensity (I1) and

the RMS surface roughness (Rq1) in response to the

tool geometry effects. The consistency is confirmed.

Fig. 10(b) plots the regression of the sum of intensity

(IT) and the RMS surface roughness (Rq). They change

identically at high feed rates, while Rq is higher than ITat low feed rates, because the peak signal with the low-

frequency intensity is not a single indicator. More than

one low-frequency vibration, in the experiment, was

identified in Sections 4.1 and 4.2 as the major influen-

cing factors at low feed rates. Errors thus occur at low

feed rates. The experimental results indicate that both

the intensity I1 and the RMS surface roughness (Rq1)

of the tool geometry effects approached zero at a feed

rate of 1 mm/min. Therefore, the measured RMS sur-

face roughness (Rq) at a feed rate of 1 mm/min was

produced only by low-frequency vibration and the

measuring instrument.Table 8 presents the RMS surface roughness mea-

sured at 1 mm/min feed rate. The average value is

0.0085 lm. The total Rq becomes

RqT ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiRq21 þ Rq22 þ Rq23

q¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiRq21 þ Rq2K

q

¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiRq21 þ ð0:0085Þ2

qð4Þ

where RqK is the RMS surface roughness that was

nearly a constant at a feed rate of 1 mm/min.

Eq. (4) can predict the RMS surface roughness and

yields results that agree with the experimental data, as

shown in Fig. 11(a).Table 9 presents the average surface roughness of

0.00665 lm measured at a feed rate of 1 mm/min. Thetotal Ra becomes

RaT ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiRa21 þ Ra22 þ Ra23

q¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiRa21 þ Ra2K

q

¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiRa21 þ ð0:00665Þ2

qð5Þ

Fig. 11(b) indicates good agreement between the pre-

diction of Eq. (5) and the experimental results.This study assumed that the tool is maintained in

good condition during machining. Fig. 12 presents an

SEM photograph of a used tool after it has cut a

length of 100 km. The tool still has a sharp nose

edge.Fig. 13 shows the quantity of a workpiece measured

by AFM. Its surface roughness is Ra ¼ 2:8 nm. Thequantity of the same workpiece measured by Talysurf

is Ra ¼ 6:5 nm. The measuring instrument is indeed anexternal factor that contributes to the intensity of sur-

face roughness, as shown in Fig. 7. The use of a precise

instrument can reduce errors.

4.4. Critical feed rate

Reducing the feed rate does not necessarily improve

surface roughness. The low-frequency vibration exists

at a low feed rate. The minimum feed rate that gen-

Fig. 9. Relationship between I2 intensity and tool nose radius.

Table 6

The ANOVA of measuring instrument intensity

Source T
ype III sum
of squares

df M
ean square F S ig.
Model 1
:791� 10�3 16 1 :119� 10�4 51.994 0 .000
D 1
:738� 10�5 9 1 :931� 10�6 0.897 0 .529
C 2
:501� 10�6 2 1 :250� 10�6 0.581 0 .560
B 7
:168� 10�6 2 3 :584� 10�6 1.665 0 .191
A 8
:417� 10�6 2 4 :208� 10�6 1.955 0 .144
Error 5
:468� 10�4 254 2 :153� 10�6 Total 2 :338� 10�3 270
A: spindle revolution; B: tool nose radius; C: cutting depth; D: feed

rate.

Table 7

Frequency and intensity of influencing factors

Influence

source

T

g

ool

eometry

Low-frequency

vibration

M

in

easuring

strument

Frequency (Hz) S
pindle rate
ðrpmÞ=60
<7 � 170
Intensity (mm s) /
feed2=noseradius
0.002–0.005 0
.002–0.004


erates the best mirror surface roughness is defined the

critical feed rate. In the experiment, the low-frequency

intensity is between 0.002 and 0.005 mm s. When the

tool geometry contribution to intensity becomes lower

than 0.005 mm s, the dominant factor shifts from the

tool geometry to the low-frequency vibration. Then,

further reduction of the feed rate is ineffective. Table 10

shows the critical feed rate under different cutting con-

ditions. Ra(cv) represents the surface roughness at the

critical feed rate, and RaK represents the surface rough-

ness at a feed rate of 1 mm/min. When the feed ratedecreases from the critical rate to 1 mm/min, surfaceroughness can be improved only by 1.7 nm, whereasthe production rate is reduced many times by reducingthe feed rate. Therefore, identifying the critical feedrate efficiently yields favorable quality and productionrate.

5. Conclusions

FFT analysis can identify the contributions of thetool geometry, environmental low-frequency vibrationsand the measuring instrument, which are the three fac-tors that dominate surface roughness in the diamondturning of a lens mold. The intensities of low-frequencyvibration and the measuring instrument are almostconstant. The tool geometry contributes to the surfaceroughness according to the model of the ideal configur-ation. It varies with feed rate, rate of spindle revolutionand tool nose radius. Based on these three major fac-tors, the RMS surface roughness model can be con-structed. The model agrees well with the experimentalresults. The analysis also yields the critical feed ratethat maximizes machining productivity; reducing the

Comparison of regressions of intensity and RMS surface roughness. (a) Tool geometry, and (b) sum of in
Fig. 10. tensity.
Table 8

RMS surface roughness measured at 1 mm/min feed rate

Revolution rate (rpm) T
ool nose radius (mm) RqK (lm)
1500 1
.02 0.0083
1000 1
.02 0.0097
800 1
.02 0.0085
1500 0
.52 0.0077
1000 0
.52 0.0076
800 0
.52 0.0074
1500 0
.23 0.0086
1000 0
.23 0.0097
800 0
.23 0.0085
Average
0.0085


rate below this value only slightly improves the surface

roughness but at the cost of severely reducing the pro-

duction rate.

Acknowledgements

The authors would like to thank the National

Science Council of the Republic of China for finan-

cially supporting this research under contract no.

NSC90-2212-E007-041.

ig. 11. Comparison of surface roughness. (a) RMS surface roughness, and (b) average surface roughness
F .
Fig. 12. SEM photograph of a tool at 100 km cutting distance.

Table 9

Average surface roughness measured at 1 mm/min feed rate

Revolution rate (rpm) T
ool nose radius (mm) RaK (lm)
1500 1
.02 0.0066
1000 1
.02 0.0078
800 1
.02 0.0067
1500 0
.52 0.0061
1000 0
.52 0.0062
800 0
.52 0.0059
1500 0
.23 0.0068
1000 0
.23 0.0078
800 0
.23 0.0067
Average
0.00665


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Table 10

Critical feed rate versus Ra(cv) and RaK

Cutting

condition

C

f

ritical

eed rate

(mm/min)

Ra(cv) (lm) R
aK (lm) RaðcvÞ � RaK
1500 rpm

1.02 mm

1
0 0.0075 0 .0066 0.0009
1000 rpm

1.02 mm

8
0.0080 0 .0078 0.0002
800 rpm

1.02 mm

5
0.0079 0 .0067 0.0012
1500 rpm

0.518 mm

1
0 0.0073 0 .0061 0.0012
1000 rpm

0.518 mm

8
0.0079 0 .0062 0.0017
800 rpm

0.518 mm

5
0.0076 0 .0059 0.0017
1500 rpm

0.23 mm

3
0.0068 0 .0068 0
1000 rpm

0.23 mm

1
0.0078 0 .0078 0
800 rpm

0.23 mm

3
0.0067 0 .0067 0


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signal analysis of surface roughness in diamond turning of lens molds

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