- Email: [email protected]
Microscopic topographical analysis of tool vibration effects on diamond turned optical surfaces
Precision Engineering Journal of the International Societies for Precision Engineering and Nanotechnology 26 (2002) 168–174
Microscopic topographical analysis of tool vibration effects on diamond turned optical surfaces Dong-Sik Kim, In-Cheol Chang, Seung-Woo Kim∗ Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology, BUPE Creative Research Initiatives Group, Science Town, Taejon 305-701, South Korea Received 30 September 2000; accepted 24 September 2001
Abstract In diamond turning for the manufacture of optical surfaces, a certain degree of relative vibration is inevitably encountered between the tool and work, deteriorating the surface quality. In this paper we first describe how the tool vibration affects the surface profiles in microscopic level, and then propose a metrological scheme to identify any existence of tool vibration with a minimum effort of surface measurement and analysis. © 2002 Elsevier Science Inc. All rights reserved. Keywords: Diamond turning; Surface analysis; Tool vibration; Topography
1. Introduction Diamond turning is widely used in the manufacture of optical flats and aspheric surfaces, in which precision surface profiles are generated with a single diamond tool of well defined tip geometry under computer numerical control of machine axes. The surface roughness of machined surface is primarily determined by the cutting conditions of tool tip radius, depth of cut, and feed rate. Another factor affecting the surface roughness is the relative vibration inevitably encountered between the tool and work, which deteriorates the surface quality. Contact type stylus instruments have long been used to inspect the level of the tool vibration appearing on the work surface [1,2] but their performance is limited as the measurement is unidirectional. Nowadays, quite a few interferometric optical surface profiling techniques providing complete 3-D surface profiles with sub-micron resolutions are available [3,4], but no systematic surface analysis methodology has been established as yet to identify diverse effects of tool vibration from the measured surface roughness data. In this investigation we first speculate a surface generation model of diamond turning that explains topological effects of tool vibration on surface profiles in microscopic topographical level. Then we propose a metrological ∗ Corresponding author. Tel.: +82-42-869-3217; fax: +82-42-869-3210. E-mail address: [email protected] (S.-W. Kim).
scheme to single out any existence of tool vibration with a minimum effort of surface measurement.
2. Surface generation model Let us assume that there is a vibration motion between the tool and work whose frequency and amplitude are f0 and A, respectively. This relative movement causes a height variation on the work surface, which may be expressed in the time domain as z(t) = A(1 − cos(2πf0 t))
(1)
In the cutting process of face turning as illustrated in Fig. 1(a), the tool moves on the work surface following the spiral trajectory that may be described in the r − θ coordinates as r =n·s·t
and
θ = 2π · n · t
(2)
where n is the rotation spindle speed given in (rev/s) and s is the feed rate of the tool in the radial direction (mm/rev). If the time variable t of Eq. (1) is replaced with the relation of t = r/ns of Eq. (2), the height variation along the radial direction is obtained as f0 r z(r) = A 1 − cos 2π (3) n s
0141-6359/02/$ – see front matter © 2002 Elsevier Science Inc. All rights reserved. PII: S 0 1 4 1 - 6 3 5 9 ( 0 1 ) 0 0 1 1 5 - 5
D.-S. Kim et al. / Precision Engineering 26 (2002) 168–174
169
Fig. 1. Tool center profile in r − θ coordinates modulated by a relative vibration between the tool and work. (a) Spiral tool locus on the work; (b) modulated surface profile in the radial direction.
Fig. 1(b) shows how the tool vibration modulates the radial profile in combination with the cutting conditions of n and s. To describe the radial profile in a more explicit form, the frequency ratio is defined as f0 =a+b n
(4)
where a is an integer while b is the fraction in the range of −0.5 to 0.5. Substituting the above definition of the frequency ratio into the harmonic term of Eq. (3) and applying trigonometric rules result in r r r cos 2π(a + b) = cos 2π a cos 2π b s s s r r − sin 2π a sin 2π b s s r (5) = cos 2π b s Note that only the discrete points where the tool center passes are considered in the above derivation so that the ratio of r/s is always integer values, yielding cos(2π a(r/s)) = 1 and sin(2π a(r/s)) = 0. It is apparent from the result of Eq. (5) that the integer value a of the frequency ratio has no effect on the radial profile produced by the tool center. On the other hand, only the fraction b modulates the radial profile with the wavelength of s λ= (6) |b|
r − θ relation of θ = 2π r/s by omitting the time variable t. Then, substituting the relation into Eq. (3) results in z(r, θ ) = A(1 − cos(aθ + bθ )) 2π = A 1 − cos aθ + sign(b) r λ
where sign(b) ≡ b/|b|. This r − θ spatial expression of generated work surface allows the contour line of zero height to be determined by solving z(r, θ ) = 0. The solution is for the case when the argument of the cosine term becomes zero, i.e., aθ + sign(b)
2π r=0 λ
or
r + sign(b)
aλ θ =0 2π (9)
The above contour line is in fact another spiral, and we call it ‘flute’ as distinguished from the spiral groove generated by the tool feed. Fig. 2 shows how the whole work surface is generated from the trajectory of the tool center that has been derived
where |b| indicates the absolute value of b. Consequently, the radial profile of Eq. (3) is rewritten as 2π z(r) = A 1 − cos r (7) λ Now, to describe the work surface profile in the r − θ coordinates, the spiral trajectory of Eq. (2) is converted to the
(8)
Fig. 2. Surface profile in consideration of tool tip radius.
170
D.-S. Kim et al. / Precision Engineering 26 (2002) 168–174
so far. If the tool tip has a radius of Rt , the height z(r, θ) of a surface point located at r from the nearest tool center path is geometrically determined as z(r + r, θ ) = z(r, θ ) + Rt − Rt 2 − r 2 (10) where −s/2 ≤ r < s/2. 3. Identification of vibration components Fig. 3 shows a representative microscopic topography of work surface graphically generated by computer simulation using the proposed surface generation model of Eqs. (9) and (10). A long fine spiral groove indicates the tool path, whose cross-section is of tool tip shape. The spiral groove undergoes sinusoidal variation in height due to the vibration between the tool and work. One interesting feature to note is that when the surface is seen in macroscopic scale, several ‘flutes’ are observed of which radial wavelength is much larger than the given feed rate. The flutes are a 3-D result of surface modulation induced by the tool vibration in combination with the spindle speed as explained in Eq. (9). The overall pattern of flutes are characterized by their number, orientation, and spacing, which are in fact determined by the integer part a and fraction b of the frequency ratio defined in Eq. (4). A series of computer simulation results presented in Fig. 4 explains in detail how the flutes are related to the frequency ratio. First, the number of the flutes is identical to
the integer part a. This fact is simply verified from Eq. (8); during the interval while θ undergoes one circular variation of 2π for a given value of r, the cosine harmonic term experiences zero a times. Second, the orientation of the flutes are decided by sign(b), which is explained from Eq. (9); when r increases following a contour line, the direction of θ increase is decided by sign(b), i.e, counterclockwise if sign(b) is positive while clockwise vise versa. Finally, the spacing of the flutes is determined by the absolute value of b, which has already been described in Eq. (6). It is also worthwhile to note two extreme cases of Fig. 4(a) and (f); the former is for a = 0 and corresponding flutes are in a concentric configuration, while the latter is for b = 0 and flutes become straight. Now our attention is given to establishing a systematic method that can identify any existence of tool vibration and subsequently quantify its frequency and amplitude. At this point it is important to point out that the industrial practice of judging the surface quality using only radial profiles usually obtained using a stylus instrument is not sufficient to identify the tool vibration because radial profiles lacks in information on the integer part a of the frequency ratio. The most reliable way is consequently to trace the height variation of the spiral groove along the tool path and perform subsequent Fourier transform analysis of the traced height data. Fig. 5(a) shows an exemplary result that has been carried out against the surface profile given in Fig. 3. This spiral surface analysis clearly reveals there exists a vibration component
Fig. 3. Simulated microscopic surface topography: feed rate s = 0.020 mm/rev, spindle speed n = 10 rps, and relative vibration frequency f0 = 21 Hz.
D.-S. Kim et al. / Precision Engineering 26 (2002) 168–174
171
Fig. 4. Variation of flutes with different frequency ratio: (a) n = 10 rev/s, f0 = 2 Hz; (b) n = 10 rev/s, f0 = 22 Hz; (c) n = 10 rev/s, f0 = 52 Hz; (d) n = 10 rev/s, f0 = 18 Hz; (e) n = 10 rev/s, f0 = 21 Hz; (f) n = 10 rev/s, f0 = 20 Hz.
of 21.0 Hz, which is in fact identical to the true value of frequency used in simulating the surface. Another approach of detecting the tool vibration is to implement the radial and circumferential surface analysis separately but in sequence to determine the frequency ratio as demonstrated in Fig. 5(b) and (c). The radial analysis provides λ = 5.0 and b = 0.1, while the circumferential analysis gives a = 2. Combining the measured values of a and b, the vibration frequency is obtained as 21.0 Hz, which is in perfect agreement with that of the spiral surface analysis. Then a question arises; what is the major merit of the latter radial–circumferential analysis in comparison with the former spiral analysis. The answer is the quantity of surface height data to be taken and processed. The spiral analysis requires 2-D N × N array data, while the radial–circumferential analysis work with two sets of
1-D data, 2N. Thus, a great reduction in measurement and computation time can be obtained.
4. Experiments and discussion The surface generation model and surface analysis methods described so far were verified through a series of experimental work. Fig. 6 shows the measurement system used to collect surface profiles from actual specimens. The measuring probe operates on the principle of phase shifting interferometry using white light [5,6]. A single measurement produces an array height data of 640 × 480 points with a lateral resolution of 3.9 m and a height resolution of 1.2 nm. To extend the measurement range in the
172
D.-S. Kim et al. / Precision Engineering 26 (2002) 168–174
Fig. 5. Identification of relative vibration frequency by simulation: feed rate s = 0.020 mm/rev, spindle speed n = 10 rps and relative vibration frequency f0 = 21 Hz. (a) Spiral direction; (b) radial direction; (c) circumferential direction.
Fig. 6. Ultra-precision 3-D measurement system for experiments.
D.-S. Kim et al. / Precision Engineering 26 (2002) 168–174
lateral dimension, ‘stitching’ technique was used which pieces together individually measured patches by minimizing the matching errors in the intentionally overlapped areas of measurement [7]. For this, the work is placed on a precision x–y stage supported by aerostatic bearings, of which lateral position is measured by heterodyne laser interferometers of 4.8 nm resolution while being moved by linear motors.
173
Fig. 7(a) shows an exemplary measurement result of an actual Al specimen machined with 0.020 mm/rev feed rate, 25 rps spindle speed, and 2.0 mm tool tip radius. The magnified view about the center of the specimen clearly reveals that there are spiral flutes generated by the modulation of the tool vibration. Fig. 7(b) shows the spiral profile that was obtained from the height variation along the tool trajectory, where a dominant frequency of 192.5 Hz is found through
Fig. 7. Identification of relative vibration frequency by experiment; feed rate s = 0.020 mm/rev, rotational speed n = 25 rps. (a) Measured surface profiles; (b) spiral direction; (c) circumferential direction; (d) radial direction.
174
D.-S. Kim et al. / Precision Engineering 26 (2002) 168–174
Table 1 Identification of tool vibration for different cutting conditions Rotational speed n (rps)
Feed rate s (mm/rev)
Radial–circumferential 1/
25.0 25.0 25.0 25.0 25.0 16.7
0.013 0.016 0.020 0.023 0.026 0.020
(mm−1 )
22.8 18.2 15.0 13.0 11.4 23.0
b
a
f0 (Hz)
0.295 0.290 0.300 0.300 0.300 0.460
8 8 8 8 8 12
192.6 192.7 192.5 192.6 192.5 192.8
192.4 192.8 192.5 192.4 192.4 192.8
192.6
192.6
Average
FFT analysis. Fig. 7(c) and (d) are the radial and circumferential profiles, respectively. The radial profile contains a modulated wavelength that provides the fractional part of the frequency ratio, i.e., b = 0.302. In addition, the circumferential profile gives the integer part of the frequency ratio as a = 8. Consequently, using Eq. (3), the vibration frequency is determined as 192.5 Hz, which is identical to the previous result obtained from the spiral analysis of Fig. 6(b). Table 1 illustrates the test results performed on six specimens prepared by the same machine as used for the previous test of Fig. 7 but with different cutting conditions. The results show that almost identical vibration frequencies are found regardless of cutting conditions, which implies that the proposed vibration identification method is reliable. 5. Conclusions We have described that the tool vibration affects the surface profiles in the form of flutes of which number, orientation, and spacing are determined by the integer and fraction of the frequency ratio of the tool vibration to the spindle speed. Then, as a systematic approach to identify the tool frequency from measured surface profiles, we have proposed two methods of spiral and radial–circumferential analysis using microscopic surface profile data obtained by phase
Spiral profile f0 (Hz)
measuring interferometry. Computer simulation and experimental results prove that our approach is well capable of identifying any existence of tool vibration with a minimum effort of surface measurement and analysis. References [1] Sato T, Li M, Takata S, Hiraoka H, Li CQ, Xing XZ, Xiao XG. Analysis of surface roughness generation in turning operation and its applications. Ann CIRP 1985;34(1):473–6. [2] Takasu S, Masuda M, Nishiguchi T, Kobayashi A. Influence study of vibration with small amplitude upon surface roughness in diamond machining. Ann CIRP 1985;34(1):463–7. [3] Weck M, Modemann K. Surface quality as a function of static and dynamic machine-tool behavior during the cutting process. Surf Topography 1988;1:291–302. [4] Weck M. Performance analysis of feed-drive systems in diamond turning by machining specified test samples. Ann CIRP 1992;41(1):601–4. [5] Kim S-W, Kim D-S, Chang I-C, Keem T-H, Yoo S-B. Very large scale phase measuring interferometry of work surfaces for diagnostic analysis of the diamond turning process. In: Proceedings of the 1st European Society for Precision Engineering and Nanotechnology. Bremen, Germany, 1999. p. 24–7. [6] Wyant JC, Creath K. Advances in interferometric optical profiling. Int J Mach Tools Manufact 1992;1/2(5–10). [7] Tang S. Stitching: high spatial resolution microsurface measurements over large areas. SPIE 1998;3479:43–9.
Microscopic topographical analysis of tool vibration effects on diamond turned optical surfaces Dong-Sik Kim, In-Cheol Chang, Seung-Woo Kim∗ Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology, BUPE Creative Research Initiatives Group, Science Town, Taejon 305-701, South Korea Received 30 September 2000; accepted 24 September 2001
Abstract In diamond turning for the manufacture of optical surfaces, a certain degree of relative vibration is inevitably encountered between the tool and work, deteriorating the surface quality. In this paper we first describe how the tool vibration affects the surface profiles in microscopic level, and then propose a metrological scheme to identify any existence of tool vibration with a minimum effort of surface measurement and analysis. © 2002 Elsevier Science Inc. All rights reserved. Keywords: Diamond turning; Surface analysis; Tool vibration; Topography
1. Introduction Diamond turning is widely used in the manufacture of optical flats and aspheric surfaces, in which precision surface profiles are generated with a single diamond tool of well defined tip geometry under computer numerical control of machine axes. The surface roughness of machined surface is primarily determined by the cutting conditions of tool tip radius, depth of cut, and feed rate. Another factor affecting the surface roughness is the relative vibration inevitably encountered between the tool and work, which deteriorates the surface quality. Contact type stylus instruments have long been used to inspect the level of the tool vibration appearing on the work surface [1,2] but their performance is limited as the measurement is unidirectional. Nowadays, quite a few interferometric optical surface profiling techniques providing complete 3-D surface profiles with sub-micron resolutions are available [3,4], but no systematic surface analysis methodology has been established as yet to identify diverse effects of tool vibration from the measured surface roughness data. In this investigation we first speculate a surface generation model of diamond turning that explains topological effects of tool vibration on surface profiles in microscopic topographical level. Then we propose a metrological ∗ Corresponding author. Tel.: +82-42-869-3217; fax: +82-42-869-3210. E-mail address: [email protected] (S.-W. Kim).
scheme to single out any existence of tool vibration with a minimum effort of surface measurement.
2. Surface generation model Let us assume that there is a vibration motion between the tool and work whose frequency and amplitude are f0 and A, respectively. This relative movement causes a height variation on the work surface, which may be expressed in the time domain as z(t) = A(1 − cos(2πf0 t))
(1)
In the cutting process of face turning as illustrated in Fig. 1(a), the tool moves on the work surface following the spiral trajectory that may be described in the r − θ coordinates as r =n·s·t
and
θ = 2π · n · t
(2)
where n is the rotation spindle speed given in (rev/s) and s is the feed rate of the tool in the radial direction (mm/rev). If the time variable t of Eq. (1) is replaced with the relation of t = r/ns of Eq. (2), the height variation along the radial direction is obtained as f0 r z(r) = A 1 − cos 2π (3) n s
0141-6359/02/$ – see front matter © 2002 Elsevier Science Inc. All rights reserved. PII: S 0 1 4 1 - 6 3 5 9 ( 0 1 ) 0 0 1 1 5 - 5
D.-S. Kim et al. / Precision Engineering 26 (2002) 168–174
169
Fig. 1. Tool center profile in r − θ coordinates modulated by a relative vibration between the tool and work. (a) Spiral tool locus on the work; (b) modulated surface profile in the radial direction.
Fig. 1(b) shows how the tool vibration modulates the radial profile in combination with the cutting conditions of n and s. To describe the radial profile in a more explicit form, the frequency ratio is defined as f0 =a+b n
(4)
where a is an integer while b is the fraction in the range of −0.5 to 0.5. Substituting the above definition of the frequency ratio into the harmonic term of Eq. (3) and applying trigonometric rules result in r r r cos 2π(a + b) = cos 2π a cos 2π b s s s r r − sin 2π a sin 2π b s s r (5) = cos 2π b s Note that only the discrete points where the tool center passes are considered in the above derivation so that the ratio of r/s is always integer values, yielding cos(2π a(r/s)) = 1 and sin(2π a(r/s)) = 0. It is apparent from the result of Eq. (5) that the integer value a of the frequency ratio has no effect on the radial profile produced by the tool center. On the other hand, only the fraction b modulates the radial profile with the wavelength of s λ= (6) |b|
r − θ relation of θ = 2π r/s by omitting the time variable t. Then, substituting the relation into Eq. (3) results in z(r, θ ) = A(1 − cos(aθ + bθ )) 2π = A 1 − cos aθ + sign(b) r λ
where sign(b) ≡ b/|b|. This r − θ spatial expression of generated work surface allows the contour line of zero height to be determined by solving z(r, θ ) = 0. The solution is for the case when the argument of the cosine term becomes zero, i.e., aθ + sign(b)
2π r=0 λ
or
r + sign(b)
aλ θ =0 2π (9)
The above contour line is in fact another spiral, and we call it ‘flute’ as distinguished from the spiral groove generated by the tool feed. Fig. 2 shows how the whole work surface is generated from the trajectory of the tool center that has been derived
where |b| indicates the absolute value of b. Consequently, the radial profile of Eq. (3) is rewritten as 2π z(r) = A 1 − cos r (7) λ Now, to describe the work surface profile in the r − θ coordinates, the spiral trajectory of Eq. (2) is converted to the
(8)
Fig. 2. Surface profile in consideration of tool tip radius.
170
D.-S. Kim et al. / Precision Engineering 26 (2002) 168–174
so far. If the tool tip has a radius of Rt , the height z(r, θ) of a surface point located at r from the nearest tool center path is geometrically determined as z(r + r, θ ) = z(r, θ ) + Rt − Rt 2 − r 2 (10) where −s/2 ≤ r < s/2. 3. Identification of vibration components Fig. 3 shows a representative microscopic topography of work surface graphically generated by computer simulation using the proposed surface generation model of Eqs. (9) and (10). A long fine spiral groove indicates the tool path, whose cross-section is of tool tip shape. The spiral groove undergoes sinusoidal variation in height due to the vibration between the tool and work. One interesting feature to note is that when the surface is seen in macroscopic scale, several ‘flutes’ are observed of which radial wavelength is much larger than the given feed rate. The flutes are a 3-D result of surface modulation induced by the tool vibration in combination with the spindle speed as explained in Eq. (9). The overall pattern of flutes are characterized by their number, orientation, and spacing, which are in fact determined by the integer part a and fraction b of the frequency ratio defined in Eq. (4). A series of computer simulation results presented in Fig. 4 explains in detail how the flutes are related to the frequency ratio. First, the number of the flutes is identical to
the integer part a. This fact is simply verified from Eq. (8); during the interval while θ undergoes one circular variation of 2π for a given value of r, the cosine harmonic term experiences zero a times. Second, the orientation of the flutes are decided by sign(b), which is explained from Eq. (9); when r increases following a contour line, the direction of θ increase is decided by sign(b), i.e, counterclockwise if sign(b) is positive while clockwise vise versa. Finally, the spacing of the flutes is determined by the absolute value of b, which has already been described in Eq. (6). It is also worthwhile to note two extreme cases of Fig. 4(a) and (f); the former is for a = 0 and corresponding flutes are in a concentric configuration, while the latter is for b = 0 and flutes become straight. Now our attention is given to establishing a systematic method that can identify any existence of tool vibration and subsequently quantify its frequency and amplitude. At this point it is important to point out that the industrial practice of judging the surface quality using only radial profiles usually obtained using a stylus instrument is not sufficient to identify the tool vibration because radial profiles lacks in information on the integer part a of the frequency ratio. The most reliable way is consequently to trace the height variation of the spiral groove along the tool path and perform subsequent Fourier transform analysis of the traced height data. Fig. 5(a) shows an exemplary result that has been carried out against the surface profile given in Fig. 3. This spiral surface analysis clearly reveals there exists a vibration component
Fig. 3. Simulated microscopic surface topography: feed rate s = 0.020 mm/rev, spindle speed n = 10 rps, and relative vibration frequency f0 = 21 Hz.
D.-S. Kim et al. / Precision Engineering 26 (2002) 168–174
171
Fig. 4. Variation of flutes with different frequency ratio: (a) n = 10 rev/s, f0 = 2 Hz; (b) n = 10 rev/s, f0 = 22 Hz; (c) n = 10 rev/s, f0 = 52 Hz; (d) n = 10 rev/s, f0 = 18 Hz; (e) n = 10 rev/s, f0 = 21 Hz; (f) n = 10 rev/s, f0 = 20 Hz.
of 21.0 Hz, which is in fact identical to the true value of frequency used in simulating the surface. Another approach of detecting the tool vibration is to implement the radial and circumferential surface analysis separately but in sequence to determine the frequency ratio as demonstrated in Fig. 5(b) and (c). The radial analysis provides λ = 5.0 and b = 0.1, while the circumferential analysis gives a = 2. Combining the measured values of a and b, the vibration frequency is obtained as 21.0 Hz, which is in perfect agreement with that of the spiral surface analysis. Then a question arises; what is the major merit of the latter radial–circumferential analysis in comparison with the former spiral analysis. The answer is the quantity of surface height data to be taken and processed. The spiral analysis requires 2-D N × N array data, while the radial–circumferential analysis work with two sets of
1-D data, 2N. Thus, a great reduction in measurement and computation time can be obtained.
4. Experiments and discussion The surface generation model and surface analysis methods described so far were verified through a series of experimental work. Fig. 6 shows the measurement system used to collect surface profiles from actual specimens. The measuring probe operates on the principle of phase shifting interferometry using white light [5,6]. A single measurement produces an array height data of 640 × 480 points with a lateral resolution of 3.9 m and a height resolution of 1.2 nm. To extend the measurement range in the
172
D.-S. Kim et al. / Precision Engineering 26 (2002) 168–174
Fig. 5. Identification of relative vibration frequency by simulation: feed rate s = 0.020 mm/rev, spindle speed n = 10 rps and relative vibration frequency f0 = 21 Hz. (a) Spiral direction; (b) radial direction; (c) circumferential direction.
Fig. 6. Ultra-precision 3-D measurement system for experiments.
D.-S. Kim et al. / Precision Engineering 26 (2002) 168–174
lateral dimension, ‘stitching’ technique was used which pieces together individually measured patches by minimizing the matching errors in the intentionally overlapped areas of measurement [7]. For this, the work is placed on a precision x–y stage supported by aerostatic bearings, of which lateral position is measured by heterodyne laser interferometers of 4.8 nm resolution while being moved by linear motors.
173
Fig. 7(a) shows an exemplary measurement result of an actual Al specimen machined with 0.020 mm/rev feed rate, 25 rps spindle speed, and 2.0 mm tool tip radius. The magnified view about the center of the specimen clearly reveals that there are spiral flutes generated by the modulation of the tool vibration. Fig. 7(b) shows the spiral profile that was obtained from the height variation along the tool trajectory, where a dominant frequency of 192.5 Hz is found through
Fig. 7. Identification of relative vibration frequency by experiment; feed rate s = 0.020 mm/rev, rotational speed n = 25 rps. (a) Measured surface profiles; (b) spiral direction; (c) circumferential direction; (d) radial direction.
174
D.-S. Kim et al. / Precision Engineering 26 (2002) 168–174
Table 1 Identification of tool vibration for different cutting conditions Rotational speed n (rps)
Feed rate s (mm/rev)
Radial–circumferential 1/
25.0 25.0 25.0 25.0 25.0 16.7
0.013 0.016 0.020 0.023 0.026 0.020
(mm−1 )
22.8 18.2 15.0 13.0 11.4 23.0
b
a
f0 (Hz)
0.295 0.290 0.300 0.300 0.300 0.460
8 8 8 8 8 12
192.6 192.7 192.5 192.6 192.5 192.8
192.4 192.8 192.5 192.4 192.4 192.8
192.6
192.6
Average
FFT analysis. Fig. 7(c) and (d) are the radial and circumferential profiles, respectively. The radial profile contains a modulated wavelength that provides the fractional part of the frequency ratio, i.e., b = 0.302. In addition, the circumferential profile gives the integer part of the frequency ratio as a = 8. Consequently, using Eq. (3), the vibration frequency is determined as 192.5 Hz, which is identical to the previous result obtained from the spiral analysis of Fig. 6(b). Table 1 illustrates the test results performed on six specimens prepared by the same machine as used for the previous test of Fig. 7 but with different cutting conditions. The results show that almost identical vibration frequencies are found regardless of cutting conditions, which implies that the proposed vibration identification method is reliable. 5. Conclusions We have described that the tool vibration affects the surface profiles in the form of flutes of which number, orientation, and spacing are determined by the integer and fraction of the frequency ratio of the tool vibration to the spindle speed. Then, as a systematic approach to identify the tool frequency from measured surface profiles, we have proposed two methods of spiral and radial–circumferential analysis using microscopic surface profile data obtained by phase
Spiral profile f0 (Hz)
measuring interferometry. Computer simulation and experimental results prove that our approach is well capable of identifying any existence of tool vibration with a minimum effort of surface measurement and analysis. References [1] Sato T, Li M, Takata S, Hiraoka H, Li CQ, Xing XZ, Xiao XG. Analysis of surface roughness generation in turning operation and its applications. Ann CIRP 1985;34(1):473–6. [2] Takasu S, Masuda M, Nishiguchi T, Kobayashi A. Influence study of vibration with small amplitude upon surface roughness in diamond machining. Ann CIRP 1985;34(1):463–7. [3] Weck M, Modemann K. Surface quality as a function of static and dynamic machine-tool behavior during the cutting process. Surf Topography 1988;1:291–302. [4] Weck M. Performance analysis of feed-drive systems in diamond turning by machining specified test samples. Ann CIRP 1992;41(1):601–4. [5] Kim S-W, Kim D-S, Chang I-C, Keem T-H, Yoo S-B. Very large scale phase measuring interferometry of work surfaces for diagnostic analysis of the diamond turning process. In: Proceedings of the 1st European Society for Precision Engineering and Nanotechnology. Bremen, Germany, 1999. p. 24–7. [6] Wyant JC, Creath K. Advances in interferometric optical profiling. Int J Mach Tools Manufact 1992;1/2(5–10). [7] Tang S. Stitching: high spatial resolution microsurface measurements over large areas. SPIE 1998;3479:43–9.